Abstract
In this work, we present a method for the design of gradient index devices for elastic waves in plates. The method allows the design of devices to control the three fundamental modes, despite the fact that their dispersion relation is managed by different elastic constants. It is shown that by means of complex graded phononic crystals and thickness variations it is possible to independently design the three refractive indexes of these waves, allowing therefore their simultaneous control. The effective medium theory required for this purpose is presented, and the method is applied to the design of the Luneburg and Maxwell lenses as well as to the design of a flat gradient index lens. Finally, numerical simulations are used to demonstrate the performance of the method in a broadband frequency region.
Highlights
We have presented a method for the design of refractive devices working simultaneously for the three fundamental Lamb modes in thin plates
The method is based on the homogenization of phononic crystal plates, studied here as finite slices of phononic crystals
The performance of the method is demonstrated by means of the design of a flat gradient index (GRIN) lens and a circular Luneburg and Maxwell lens working simultaneously for the three modes
Summary
Yabin Jin[1,2], Daniel Torrent[3], Yan Pennec[2], Yongdong Pan1 & Bahram Djafari-Rouhani[2] received: 06 December 2015 accepted: 30 March 2016 Published: 14 April 2016. The major drawback for the realization of these devices for elastic waves is that, unlike acoustic waves in fluids, the propagation of elastic waves, either in bulk materials or plates, presents three polarizations, which propagate at different speeds. The authors have recently developed a design method which allows the simultaneous control of two of the three fundamental plate modes[23], namely the anti-symmetric (A0) and symmetric (S0) Lamb modes This simultaneous control was based on an effective medium theory developed for the A0 mode[24] working with Kirchhoff equation[25], which is a two-dimensional equation describing the propagation of flexural waves in thin plates. Numerical simulations by COMSOL support their functionality in a broadband frequency region
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