<p>Inspired from photonic crystals in solid state physics, the study of periodic mechanical structures known as phononic crystals or metamaterials has developed into an important research vertical in the field of dynamics and vibration. In recent years, topological insulators have taken a center stage in metamaterial and phononic crystal research due their special wave properties that offer immunity against defects and imperfections, no wave back-scattering, as well as unidirectional wave transmission. Analogous to electronic systems, topological states such as Quantum Hall Effect, Quantum Spin Hall Effect, Quantum Valley Hall Effect, etc. have been observed in mechanical systems when certain conditions and symmetries are present. More recently, higher order topological insulators have been discovered, offering unique ability to exhibit in-gap zero-dimensional corner modes which goes beyond regular one-dimensional edge states in other topological systems. Multiple studies show existence of higher order topological phases in elastic/acoustic regular Kagome lattices. While rich in dynamics, the corner modes in elastic twisted Kagome lattices have not been previously studied in literature. It is, therefore, the aim of this study to demonstrate the existence of corner modes in twisted Kagome lattices. Twisted Kagome lattices fall under the special class of Maxwell lattices, which have been at the heart of recent developments in the field of topological mechanics. The lattice is made up periodic arrangement of equilateral triangular truss elements where each triangular element is connected to three neighboring elements via perfect hinges (rotate freely) at their vertices. The angle between any two triangles is known as twist angle which characterizes the dynamic behavior of overall structure. The structure exhibit unique behavior at a critical twist angle for which the corresponding sides of neighboring triangles become perpendicular to each other. At this critical angle, called self-dual point, the dispersion band structure of the lattice becomes two-fold degenerate meaning that the dispersion branches appear in pairs of two with identical copies lying on top of each other. Moreover, the dispersion characteristics of lattices at twist angles equidistant from self-dual point are identical (dual of each other) despite their completely different physical structure. We demonstrate the existence of corner modes in the first bandgap of twisted Kagome lattice using finite parallelogram and hexagon-shaped lattices. We found that the corner modes appear only at special corners of finite lattices with 120<sup>o</sup> internal angle which lie on plane of mirror symmetry. Thus, a regular hexagon-shaped lattice exhibit six corner modes whereas parallelogram shaped lattice show only two corner modes. The corner modes appear all at the same frequency in case of self-dual Kagome lattice and they show dual behavior away from self-dual point. The corner modes observed in case of twisted Kagome lattices are reminiscent with the second order topologically protected states in acoustic and elastic regular Kagome lattices which rely on non-trivial bulk topology characterized by quantized Wannier centers.&nbsp; To this end, we also explore the analogous electronic system to explain the existence of corner modes via bulk polarization and Wannier centers. We further demonstrate the robustness of these corner modes against bulk defects and systematic perturbations. Such peculiar dynamics of self-dual Kagome lattices may find novel applications in energy harvesting, sensing, and wave localization.</p>

Corner Modes in Elastic Twisted Kagome Lattices

Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2021/volume/85543

ASME 2021 International Mechanical Engineering Congress and Exposition (IMECE2021)

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