The regularity of earthquakes, their destructive power, and the nuisance of ground vibration in urban environments, all motivate designs of defence structures to lessen the impact of seismic and ground vibration waves on buildings. Low frequency waves, in the range 1-10 Hz for earthquakes and up to a few tens of Hz for vibrations generated by human activities, cause a large amount of damage, or inconvenience; depending on the geological conditions they can travel considerable distances and may match the resonant fundamental frequency of buildings. The ultimate aim of any seismic metamaterial, or any other seismic shield, is to protect over this entire range of frequencies; the long wavelengths involved, and low frequency, have meant this has been unachievable to date. Notably this is scalable and the effects also hold for smaller devices in ultrasonics. There are three approaches to obtaining shielding effects: bragg scattering, locally resonant sub-wavelength inclusions and zerofrequency stop-band media. The former two have been explored, but the latter has not and is examined here. Elastic flexural waves, applicable in the mechanical vibrations of thin elastic plates, can be designed to have a broad zero-frequency stop-band using a periodic array of very small clamped circles. Inspired by this experimental and theoretical observation, all be it in a situation far removed from seismic waves, we demonstrate that it is possible to achieve elastic surface (Rayleigh) wave reflectors at very large wavelengths in structured soils modelled as a fully elastic layer periodically clamped to bedrock. We identify zero frequency stop-bands that only exist in the limit of columns of concrete clamped at their base to the bedrock. In a realistic configuration of a sedimentary basin 15 m deep we observe a zero frequency stop-band covering a broad frequency range of 0-30 Hz. and horizontally polarized Love waves), pressure bulk waves and shear bulk waves; surface waves cause the majority of any damage and travel farthest, but bulk pressure, and shear, waves also cause damage, especially where wave trapping occurs in sedimentary basins (so-called seismic site effects). Designing a defence structure to prevent seismic waves from reaching buildings is therefore of substantial interest particularly for long waves with frequency in the range 1-10Hz as this corresponds to the resonant fundamental frequency of many manmade structures [3]. It is, of course, these long, low frequency, waves that are the hardest to develop protection measures against and it is an open problem to develop such devices.Since the late 1980s, in optics, researchers have taken advantage of technological improvements in structuring matter to achieve control over the flow of light [4,5]. The photonic crystals typically used are periodic man-made structures that inhibit emission due to band gaps [6, 7], i.e., ranges of frequency in which light cannot propagate through the structure. The existence of stop-bands is well predicted by Floquet-Bloch theory, which can be...

We consider microstructured thin elastic plates that have an underlying periodic structure, and develop an asymptotic continuum model that captures the essential microstructural behaviour entirely in a macroscale setting. The asymptotics are based upon a two-scale approach and are valid even at high frequencies when the wavelength and microscale length are of the same order. The general theory is illustrated via one-and two-dimensional model problems that have zero-frequency stop bands that preclude conventional averaging and homogenization theories. Localized defect modes created by material variations are also modelled using the theory and compared with numerical simulations.

Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.