Small-World Disordered Lattices: Spectral Gaps and Diffusive Transport

TL;DR

This paper explores how introducing non-local connections in small-world lattices creates spectral gaps and alters vibrational transport, revealing potential for novel metamaterial functionalities like frequency filtering and stress mitigation.

Contribution

It demonstrates the emergence of spectral gaps and transition to diffusive transport in disordered small-world lattices, highlighting new ways to control wave propagation in metamaterials.

Findings

Spectral gaps appear with increasing disorder, especially in low-connectivity lattices.

Disordered lattices transition from ballistic to diffusive or super-diffusive transport.

Disorder enables frequency filtering and stress mitigation applications.

Abstract

We investigate the dynamic behavior of lattices with disorder introduced through non-local network connections. Inspired by the Watts-Strogatz small-world model, we employ a single parameter to determine the probability of local connections being re-wired, and to induce transitions between regular and disordered lattices. These connections are added as non-local springs to underlying periodic one-dimensional (1D) and two-dimensional (2D) square, triangular and hexagonal lattices. Eigenmode computations illustrate the emergence of spectral gaps in various representative lattices for increasing degrees of disorder. These gaps manifest themselves as frequency ranges where the modal density goes to zero, or that are populated only by localized modes. In both cases, we observe low transmission levels of vibrations across the lattice. Overall, we find that these gaps are more pronounced for lattice topologies with lower connectivity, such as the 1D lattice or the 2D hexagonal lattice. We then illustrate that the disordered lattices undergo transitions from ballistic to super-diffusive or diffusive transport for increasing levels of disorder. These properties, illustrated through numerical simulations, unveil the potential for disorder in the form of non-local connections to enable additional functionalities for metamaterials. These include the occurrence of disorder-induced spectral gaps, which is relevant to frequency filtering devices, as well as the possibility to induce diffusive-type transport which does not occur in regular periodic materials, and that may find applications in dynamic stress mitigation.