Simultaneous control of the S and A Lamb modes by graded phononic crystal plates

We propose a theory of gradient index devices in plates that allow the simultaneous control of both S0 and A0 Lamb modes. This is in contrast to the existing approaches that are mainly limited to the manipulation of only the lowest A0 modes. These devices are based on phononic crystal plates, which are studied in the low frequency (homogenization) limit. We demonstrate a direct relationship between the dispersion relation of these two modes in phononic crystal plates that, together with the thickness dependence of the dispersion relation of the A0 mode, allows their simultaneous control. As a matter of illustration, a flat gradient index lens and a circular Luneburg lens are designed by means of the simultaneous variation of the inclusions' radii and the thickness of the plate. Numerical simulations show that the performance of these devices is good for the two modes in a broadband frequency region and that this approach can be used to design more advanced refractive devices for the total control of guided vibrational modes. Potential applications for the detection of small vibrations and energy harvesting devices are finally discussed.