Gradient index lenses for flexural waves based on thickness variations

Abstract

This work presents a method for the realization of gradient index devices for flexural waves in thin plates. Unlike recent approaches based on phononic crystals, the present approach is based on the thickness-dependence of the dispersion relation of flexural waves, which is used to create gradient index devices by means of local variations of the plate's thickness. Numerical simulations of known circularly symmetrical gradient index lenses have been performed. These simulations have been done using the multilayer multiple scattering method and the results prove their broadband efficiency and omnidirectional properties. Finally, finite element simulations employing the full three-dimensional elasticity equations also support the validity of the designed approach. V C 2014 AIP Publishing LLC. [http://dx

Omnidirectional refractive devices for flexural waves based on graded phononic crystals

“…It is clear from equation (4) that the refractive index can be properly designed by means of the material's properties, as usual, but also by means of the plate's thickness. This last idea has been exploited by krylov et al [18] and Climente et al [19,20] for the creation of gradient index devices by means of thickness variations. From the practical point of view, the fabrication of thin plates with a region of thickness variation following a specific profile can be complex under certain conditions, specially at the micro or nano-scales.…”

“…In a recent publication [20] Climente et al proposed a set of omnidirectional refractive lenses for flexural waves based on thickness variations. This is possible given that the propagation velocity of flexural waves in thin plates depends not only on the material's parameters, but also on the plate's thickness.…”

Effective medium theory for elastic metamaterials in thin elastic plates

“…For instance, cloaking devices, [14][15][16] flat lenses based on negative refraction, 17,18 gradient index flat and circular lenses [19][20][21] and omnidirectional absorbers 22,23 for flexural waves have been recently proposed and experimentally verified. Also, the dispersion relation of plates with periodic arrangements of rigid pins, 24,25 holes, 26 attached pillars [27][28][29] or point-like spring-mass resonators [30][31][32] attached to them has been investigated by several groups, as well as the resonant properties of complex inclusions.…”

Experiments on Maxwell's fish-eye dynamics in elastic plates

“…1. It has indeed been suggested that plates of varying thicknesses make focussing effects of Lamb waves possible [17] and this provides a nice playground for transformation platonics.…”

“…We stress that analysis of time focusing of Lamb waves can also help to solve the controversy on whether or not Maxwell's fish-eye makes a perfect lens [11,13,14], but this is beyond the scope of the present work as it requires a plate with a circular stressfree boundary. Moreover, other thickness variations can be envisaged [17], what makes for instance Eaton lenses for flexural waves possible. We note that such lensing effects might be observed with pillar based metamaterial plates [24] with a variation of pillars's height like in Eq.…”