Seismic invisibility: elastic wave cloaking via symmetrized transformation media

Abstract

Transformation media theory, which steers waves in solids via an effective geometry induced by a refractive material (Fermat's principle of least action), provides a means of controlling vibrations and elastic waves beyond the traditional dissipative structures regime. In particular, it could be used to create an elastic wave cloak, shielding an interior region against elastic waves while simultaneously preventing scattering in the outside domain. However, as a true elastic wave cloak would generally require materials with stiffness tensors lacking the minor symmetry (implying asymmetric stress), the utility of such an elastic wave cloak has thus far been limited by the challenge of fabricating these materials. Here we develop a means of overcoming this limitation via the development of a symmetrized elastic cloak (SEC), sacrificing some of the performance of the perfect cloak for the sake of restoring the minor symmetry. We test the performance of the SEC for shielding a tunnel against seismic waves, showing that it can be used to reduce the average displacement within the tunnel by an order of magnitude (and reduce energy by two orders of magnitude) for waves above a critical frequency of the cloak. This critical frequency, which corresponds to the generation of surface waves at the cloak-interior interface, can be used to develop a simple heuristic model of the SEC's performance for a generic problem.The need to protect objects against unwanted mechanical vibration and wave incidence is a long-standing subject in engineering [1,2]. The problem arises in multiple scenarios including blocking sound/elastic waves and eliminating unwanted mechanical resonances [1, 2] (vibration isolation) or guarding against nonlinear mechanical shock waves and preventing the collapse of structures under incident seismic waves [3, 4] (earthquake engineering). Techniques to accomplish these goals have included strengthening structures, modifying structural resonances through additive engineering or anti-resonance, developing flexible structures that can withstand large deformations, or including dissipative elements [1][2][3][4]. Recently, techniques from phononics and acoustic wave engineering have been adapted to vibration isolation/earthquake engineering (VIEE), most prominently by the inclusion of phononic/sonic crystals as a means of shielding against seismic surface waves [1, 4-11] (which contain a much smaller portion of energy from an earthquake than bulk waves, particularly shear waves [12]). However, one limitation of all these techniques is that they can only seek to mitigate elastic waves, reducing the local amplitude of the earthquake or allowing a structure to withstand an unmodified vibration. In addition, many of these techniques have focused upon narrow frequency bands (e.g. modifying resonances), particularly for low frequency applications. While these are primary concerns, minimization of broadband or higher frequency transmissions to a structure is of increasing engineering interest nowadays as the reliance on e...

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Modified structural Luneburg lens for broadband focusing and collimation

“…Recently, phononics crystals and acoustic metamaterials have been exploited for vibration isolation, earthquake engineering, and flexural wave cloaking. Most of these efforts have been focused on the development of periodic cloaking structures or transformation acoustic approaches [25][26][27][28][29][30][31][32][33][34]. However, periodic structures have the limitation of narrow band [35,36] and the transformation acoustics method requires complicated design and stringent refractive indices [37,38].…”

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