On the band gap universality multiphase laminates and its applications

TL;DR

This paper extends the universal manifold concept for band structures in multiphase laminates, enabling analysis of gap density, width, and tunability, with applications to nonlinear materials and optimization.

Contribution

It introduces higher-dimensional manifolds encapsulating multiphase laminate band structures and demonstrates their use in analyzing gap properties and tunability.

Findings

Gap density is invariant under certain properties.

Manifolds facilitate optimization of gap widths.

Band structure tunability via pre-deformations in nonlinear laminates.

Abstract

\citet{Shmuel2016JMPS} discovered that all infinite band structures of waves at normal incidence in two-phase laminates are encapsulated in a compact universal manifold. We show that manifolds of higher dimensionality encapsulate the band structures of all multiphase laminates. We use these manifolds to determine the density of gaps in the spectrum, and prove it is invariant with respect to certain properties. We further demonstrate that these manifolds are useful for formulating optimization problems on the gaps width, for which we develop a simple bound. Using our theory, we numerically study the dependency of the gaps density and width on the impedance and number of phases. Finally, we show that in certain settings, our analysis applies to non-linear multiphase laminates, whose band diagram is tunable by finite pre-deformations. Through simple examples, we demonstrate how the universality of our representation is useful for characterizing this tunability in multiphase laminates.