High directivity and confinement of flexural waves through ultra-refraction in thin perforated plates

We propose a design of a periodically perforated thin plate leading to omni-directivity and confinement of bending waves via ultra-refraction. We first analyze the band spectrum of the bi-harmonic operator for an array of freely vibrating square voids arranged in a checkerboard fashion using finite elements: the spectrum displays nearly flat bands associated with slow flexural waves shown to be associated with resonances of square cells with stress-free boundary conditions. We find some point in the reciprocal space where the curvature of the dispersion diagram vanishes. We then demonstrate that a concentrated point force generating a bending wave within a finite array of 221 perforations displays an enhanced local density of states and radiates a flexural wave mostly in the direction perpendicular to the checkerboard slab. We next propose a simple reconstruction of the effective density associated with the platonic crystal by comparison of local density of states. This also allows us to quantify the confinement of elastic energy with the platonic crystal. We finally produce the emission diagram which demonstrates that the radiative device acts as an omni-directive super-antenna. Copyright (C) EPLA, 2010