A tunable electromagnetic metagrating

We explore electromagnetic (EM) wave incidence upon gratings of reconfigurable metamaterial cylinders, which collectively act as a metagrating, to identify their potential as reconfigurable subwavelength surfaces. The metacylinders are created by a closely spaced, microstructured array of thin plates that, in the limit of small inter-plate spacing, are described by a semi-analytical continuum model. We build upon metacylinder analysis in water waves, translating this to EM for TE polarization (longitudinal magnetic field) for which the metacylinders exhibit anisotropic scattering; this is exploited for the multiple scattering of light by an infinite metagrating of uniform cylinder radius and angle, for which we retrieve the far-field reflection and transmission spectra for plane-wave incidence. These spectra reveal unusual effects including perfect reflection and a negative Goos–Hänchen shift in the transmitted field, as well as perfect symmetry in the far-field scattering coefficients. The metagrating also hosts Rayleigh–Bloch surface waves, whose dispersion is contingent on the uniform cylinder angle, shifting under rotation towards the light-line as the cylinder angle approaches the horizontal. For both plane-wave scattering and the calculation of the array-guided modes, the cylinder angle is the principal variable in determining the wave interaction, and the metagrating is tunable simply through rotation of the constituent metacylinders.

Clusters of Bloch waves in three-dimensional periodic media

We consider acoustic wave propagation through a periodic array of the inclusions of arbitrary shape. The inclusion size is much smaller than the array period while the wavelength is fixed. We derive and rigorously justify the dispersion relation for general frequencies and show that there are exceptional frequencies for which the solution is a cluster of waves propagating in different directions with different frequencies so that the dispersion relation cannot be defined uniquely. Examples are provided for the spherical inclusions.

Dispersive and effective properties of two-dimensional periodic media

We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions for the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the cell size is small compared with the wavelength, but large compared with the radius a of the inclusions. Explicit formulae are obtained for asymptotic expansion of the solution of the problem in terms of the dimensionless magnitude q of the wavevector and radius a. This leads to explicit formulae for the effective dielectric tensor and the dispersion relation with the rigorously justified error of order O((q ² + a ²)^5/2).

An integral equation method for the homogenization of unidirectional fibre-reinforced media; antiplane elasticity and other potential problems

In Parnell & Abrahams (2008 Proc. R. Soc. A464, 1461–1482. (doi:10.1098/rspa.2007.0254)), a homogenization scheme was developed that gave rise to explicit forms for the effective antiplane shear moduli of a periodic unidirectional fibre-reinforced medium where fibres have non-circular cross section. The explicit expressions are rational functions in the volume fraction. In that scheme, a (non-dilute) approximation was invoked to determine leading-order expressions. Agreement with existing methods was shown to be good except at very high volume fractions. Here, the theory is extended in order to determine higher-order terms in the expansion. Explicit expressions for effective properties can be derived for fibres with non-circular cross section, without recourse to numerical methods. Terms appearing in the expressions are identified as being associated with the lattice geometry of the periodic fibre distribution, fibre cross-sectional shape and host/fibre material properties. Results are derived in the context of antiplane elasticity but the analogy with the potential problem illustrates the broad applicability of the method to, e.g. thermal, electrostatic and magnetostatic problems. The efficacy of the scheme is illustrated by comparison with the well-established method of asymptotic homogenization where for fibres of general cross section, the associated cell problem must be solved by some computational scheme.

Vortex-type elastic structured media and dynamic shielding

The paper presents an approach to modelling a novel elastic metamaterial structure that possesses non-trivial dispersion features. A system of spinners has been embedded into a two-dimensional periodic lattice system. The analysis of the motion of the spinners is used to derive an expression for a ‘chiral term’ in the equations describing the dynamics of the lattice. Dispersion of elastic waves is shown to possess innovative filtering and polarization properties induced by the vortex-type nature of the structured media. The related effective behaviour in a continuous medium is implemented to build a shielding cloak around an obstacle. Analytical work is accompanied by numerical illustrations.

Propagation of elastic waves through a lattice of cylindrical cavities

We consider wave propagation through a doubly periodic array of cavity cylinders in an isotropic elastic medium using the method of matched asymptotic expansions based on the assumptions that the scatterer size is much smaller than both the wavelength and the array periodicity. There is no restriction on the wavelength relative to the periodicity, and hence the method yields explicit approximations that describe the phenomena associated with periodic media. This is illustrated with square and hexagonal lattices.

Elastic waves in arrays of elliptic inclusions

We consider in-plane elastic waves propagating through a doubly periodic array of cylinders of Tantalum (with both circular and elliptical cross-sections) which are embedded in a matrix of fused silica. We find some sonic gap for fairly small filling fractions of the cylinders which eventually vanish in the limit of high-filling fraction. In the case of a doubly periodic array of elliptical cylinders, removal of a cylinder within a macro-cell leads to two localised eigenstates.

A new integral equation approach to elastodynamic homogenization

A new theory of elastodynamic homogenization is proposed, which exploits the integral equation form of Navier's equations and relationships between length scales within composite media. The scheme is introduced by focusing on its leading-order approximation for orthotropic, periodic fibre-reinforced media where fibres have arbitrary cross-sectional shape. The methodology is general but here it is shown for horizontally polarized shear (SH) wave propagation for ease of exposition. The resulting effective properties are shown to possess rich structure in that four terms account separately for the physical detail of the composite (associated with fibre cross-sectional shape, elastic properties, lattice geometry and volume fraction). In particular, the appropriate component of Eshelby's tensor arises naturally in order to deal with the shape of the fibre cross section. Results are plotted for circular fibres and compared with extant methods, including the method of asymptotic homogenization. The leading-order scheme is shown to be in excellent agreement even for relatively high volume fractions.

Higher order asymptotic homogenization and wave propagation in periodic composite materials

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.