Biharmonic Split Ring Resonator Metamaterial: Artificially dispersive effective density in thin periodically perforated plates

We present in this paper a theoretical and numerical analysis of bending waves localized on the boundary of a platonic crystal whose building blocks are split ring resonators (SRR). We first derive the homogenized parameters of the structured plate using a three-scale asymptotic expansion in the linearized biharmonic equation. In the limit when the wavelength of the bending wave is much larger than the typical heterogeneity size of the platonic crystal, we show that it behaves as an artificial plate with an anisotropic effective Young modulus and a dispersive effective mass density. We then analyze dispersion diagrams associated with bending waves propagating within an infinite array of SRR, for which eigen-solutions are sought in the form of Floquet-Bloch waves. We finally demonstrate that this structure displays the hallmarks of All-Angle-Negative-Refraction(AANR) and it leads to superlensing and ultrarefraction effects, interpreted thanks to our homogenization model as a consequence of negative and vanishing effective density, respectively.Comment: 17 pages, 6 figure

Transformation seismology: composite soil lenses for steering surface elastic Rayleigh waves.

Metamaterials are artificially structured media that exibit properties beyond those usually encountered in nature. Typically they are developed for electromagnetic waves at millimetric down to nanometric scales, or for acoustics, at centimeter scales. By applying ideas from transformation optics we can steer Rayleigh-surface waves that are solutions of the vector Navier equations of elastodynamics. As a paradigm of the conformal geophysics that we are creating, we design a square arrangement of Luneburg lenses to reroute Rayleigh waves around a building with the dual aim of protection and minimizing the effect on the wavefront (cloaking). To show that this is practically realisable we deliberately choose to use material parameters readily available and this metalens consists of a composite soil structured with buried pillars made of softer material. The regular lattice of inclusions is homogenized to give an effective material with a radially varying velocity profile and hence varying the refractive index of the lens. We develop the theory and then use full 3D numerical simulations to conclusively demonstrate, at frequencies of seismological relevance 3–10 Hz, and for low-speed sedimentary soil (v(s): 300–500 m/s), that the vibration of a structure is reduced by up to 6 dB at its resonance frequency

Achieving control of in-plane elastic waves

We derive the elastic properties of a cylindrical cloak for in-plane coupled shear and pressure waves. The cloak is characterized by a rank 4 elasticity tensor with 16 spatially varying entries which are deduced from a geometric transform. Remarkably, the Navier equations retain their form under this transform, which is generally untrue [Milton et al., New J. Phys. 8, 248 (2006)]. We numerically check that clamped and freely vibrating obstacles located inside the neutral region are cloaked disrespectful of the frequency and the polarization of an incoming elastic wave.Comment: 9 pages, 4 figure

Transformation design of in-plane elastic cylindrical cloaks, concentrators and lenses

We analyse the elastic properties of a class of cylindrical cloaks deduced from linear geometric transforms x -> x ' in the framework of the Milton-Briane-Willis cloaking theory [New Journal of Physics 8, 248, 2006]. More precisely, we assume that the mapping between displacement fields u(x) -> u '(x ') is such that u '(x ') = A-tu(x), where A is either the transformation gradient Fij = 8x ' i/8xj or the second order identity tensor I. The nature of the cloaks under review can be three-fold: some of them are neutral for a source located a couple of wavelengths away; others lead to either a mirage effect or a field confinement when the source is located inside the concealment region or within their coated region (some act as elastic concentrators squeezing the wavelength of a pressure or shear polarized incident plane wave in their core); the last category of cloaks is classified as an elastic counterpart of electromagnetic perfect cylindrical lenses. The former two categories require either rank-4 elastic tensor and rank-2 density tensor and additional rank-3 and 2 positive definite tensors (A = F) or a rank-4 elasticity tensor and a scalar density (A = I) with spatially varying positive values. However, the latter example further requires that all rank-4, 3 and 2 tensors be negative definite (A = F) or that the elasticity tensor be negative definite (and non fully symmetric) as well as a negative scalar density (A = I). We provide some illustrative numerical examples with the Finite Element package Comsol Multiphysics when A is the identity

Non-singular arbitrary cloaks dressing three-dimensional anisotropic obstacles

We design three dimensional electromagnetic cloaks, starting from a small region of complex shape instead of a point. We derive the expression of a transformation matrix describing an objet with a surface of revolution and its associated non-singular cloak. We note that while none of the eigenvalues vanish inside the cloak, they suffer a discontinuity on its inner surface. Moreover, all three eigenvalues are independent upon the radius in the concealed object. The validity of our analytical results is confirmed by finite edge-elements computations showing scattering is much reduced when the object is dressed with the cloak. We note that neither the object nor the cloak are invisible on their own.Comment: 7 pages, 4 figure