On the band gap universality of multiphase laminates and its applications

Abstract

The band structure of a periodic medium describes which wave frequencies, termed gaps, it filters out, depending on the medium composition. Shmuel and Band (2016) discovered that all infinite band structures of two-phase laminates impinged by normal waves are remarkably encapsulated in a finite geometric object, independently of the specific laminate composition. We here unveil a generalized object that encapsulates the band structures of all multiphase laminates impinged by normal waves. The merit of such a universal object is more than mathematical beauty—it establishes a platform for unprecedented characterization of the band structure. We specifically exploit it to rigorously determine the density of the gaps in the spectrum, and prove it exhibits universal features. We further utilize it to formulate optimization problems on the gap width and develop a simple bound. Using this framework, we numerically study the dependency of the gap density and width on the impedance and number of phases. In certain settings, our analysis applies to non-linear multiphase laminates, whose band diagram is tunable by pre-deformations. Through simple examples, we demonstrate how the universal object is useful for tunability characterization. Our insights may establish a step towards engineering filtering devices according to desired spectral properties.