The topological dynamics of continuum lattice grid structures

TL;DR

This paper develops a theoretical framework to identify and analyze topological edge and corner modes in continuum lattice grid structures, revealing their potential for advanced structural applications.

Contribution

It introduces a rigorous method to determine topological phases and edge states in continuum lattice structures, applicable across various configurations and frequency ranges.

Findings

Infinite topological edge states within bandgaps identified

Analytical expressions for topological phases derived

Topological modes occur at multiple frequencies

Abstract

Continuum lattice grid structures which consist of joined elastic beams subject to flexural deformations are ubiquitous. In this work, we establish a theoretical framework of the topological dynamics of continuum lattice grid structures, and discover the topological edge and corner modes in these structures. We rigorously identify the infinitely many topological edge states within the bandgaps via a theorem, with a clear criterion for the infinite number of topological phase transitions. Then, we obtain analytical expressions for the topological phases of bulk bands, and propose a topological index related to the topological phases that determines the existence of the edge states. The theoretical approach is directly applicable to a broad range of continuum lattice grid structures including bridge-like frames, square frames, kagome frames, continuous beams on elastic springs. The frequencies of the topological modes are precisely obtained, applicable to all the bands from low- to high-frequencies. Continuum lattice grid structures serve as excellent platforms for exploring various kinds of topological phases and demonstrating the topological modes at multiple frequencies on demand. Their topological dynamics has significant implications in safety assessment, structural health monitoring, and energy harvesting.