Coiled Phononic Crystal with Periodic Rotational Locking: Subwavelength Bragg Band Gaps

Abstract

Phononic crystals (PnC) are spatially periodic materials with band gaps that form by Bragg scattering of elastic waves. Within the frequency range of a band gap, wave propagation is not admitted. A long-standing limitation of this class of materials is that the wavelength for band-gap formation must be on the order of the unit-cell size. This restricts the presence of band gaps to relatively high frequencies for a given lattice spacing. Locally resonant metamaterials, on the other hand, enable the opening of low-frequency, subwavelength band gaps through resonance hybridization. However, their band gaps are characteristically narrow and require large or massive local resonators to form. Here, we break both limitations using beam-based PnCs by (1) locking the rotation degree of freedom at the edges of the primitive unit cell, and (2) coiling the PnC by applying full beam-axis rotations at the locked locations. These respective kinematic and geometric transformations convert a conventional beam PnC from its extended form with a nominal lattice constant to an extremely compact coiled configuration with a greatly reduced lattice constant. With the periodic rotational locking, the band gaps remain intact and are still large, and in fact increase noticeably in size. With the subsequent coiling, the band gaps remain based on Bragg scattering and are quantitatively conserved except now appearing at lower frequencies as dictated by the ratio of the extended-to-coiled lattice constants. This ratio defines a coiling factor, which is a measure of the reduction in the PnC unit-cell length in the direction of wave transmission while maintaining the band structure of its original extended form except for the favorable changes induced by the periodic rotational locking. A coiling factor of $\ensuremath{\beta}$ lowers, by construction, the location of the normalized central frequency of any given band gap by a factor of $\ensuremath{\beta}$. The only limitation is the need for lateral space to accommodate the coiling of the beam segments. The vibration behavior of a finite version of the coiled structure is experimentally tested demonstrating a matching band-gap response, despite the reduction in length, to that obtained by finite-element analysis of the extended rotationally locked version. This concept creates effectively subwavelength Bragg band gaps. It clears the path for PnCs to serve in applications that are orders-of-magnitude smaller in scale than are currently possible, while featuring band gaps that are significantly larger than those generated by locally resonant metamaterials.