COMBINED SEMINAR: 1/ Metamaterials: From control of waves to diffusion processes 2/High frequency homogenization: Connecting the Microstructure to the Macroscale

Abstract:
 
It has been recently suggested that structured media can bend light trajectories around an object, making it invisible, or even bending light the wrong way, making a flat lens convergent. Such electromagnetic metamaterials introduced by Sir John Pendry during the last decade have counterparts in acoustics [1], such as invisibility cloaks for surface water waves. In this talk, I will focus my attention on another type of surface waves: flexural (Lamb) waves propagating within thin elastic metamaterial plates which have been experimentally studied this year by research groups in Karlsruhe, Marseille and Paris. Interestingly, control of flexural waves opens a route towards seismic metamaterials. First experiments performed by multinational company Menard this year near the French cities of Grenoble and Lyon demonstrate that one can shield or even focus a seismic wave of magnitude 4 on the Richter scale through an array of meter sized holes drilled in a rather homogeneous soil. Finally, I will discuss some possible extension of the physics of metamaterials to thermodynamics. All of these metamaterials offer a nice playground for the high frequency homogenization (HFH) method, since much of the physics contained therein cannot be captured by classical homogenization theories in contrast with HFH.
 
[1] R. Craster, S. Guenneau, Acoustic Metamaterials: Negative refraction, imaging, lensing and cloaking, Springer Verlag, January 2013

Abstract:
 
It is highly desirable to be able to create continuum equations that embed a known microstructure through effective or averaged quantities such as wavespeeds or shear moduli. The methodology for achieving this at low frequencies and for waves long relative to a  microstructure is well-known and such static or quasi-static theories are well developed. However, at high frequencies the multiple scattering by the elements of the microstructure, which is  now of a similar scale to the wavelength, has apparently prohibited any homogenization theory. Many interesting features of, say, periodic media: band gaps, localization etc occur at frequencies inaccessible to averaging theories. The materials exhibit effective anisotropy and this leads to topical effects such as cloaking/ invisibility, flat lensing, negative refraction and to inducing directional behaviour of the waves within a structure.

Recently we have developed an asymptotic approach that overcomes this limitation, and continuum equations are developed, even though the microstructure and wavelength are now of the same order. The general theory will be described and applications to continuum, discrete and frame lattice structures will be outlined. The results and methodology are confirmed versus various illustrative exact/ numerical calculations showing that theory captures, for instance, all angle negative refraction, ultra refraction and localised defect modes.