Higher-order topological corner states and edge states in grid-like frames
Continuum grid-like frames composed of rigidly jointed beams are classic subjects in the field of structural mechanics, whose topological dynamical properties have only recently been revealed. For two-dimensional frames, higher-order topological phenomena may occur, with frequency ranges of topological states and bulk bands becoming overlapped, leading to hybrid mode shapes. Concise theoretical results are necessary to identify the topological modes in such planar continuum systems with complex spectra. In this work, we present analytical expressions for the frequencies of higher-order topological corner states, edge states, and bulk states in kagome frames and square frames, as well as the criteria of existence of these topological states and patterns of their distribution in the spectrum. We identify the edge and corner states even under their degeneracy with the bulk bands. We show that the corner states are within the bandgaps of edge states unless topological transitions occur, and demonstrate the robustness of higher-order topological states under perturbations. These theoretical results fully demonstrate that the grid-like frames, despite being a large class of two-dimensional continuum systems, have topological states that can be accurately characterized through concise analytical expressions. This work contributes to the study of topological mechanics, and the accurate and concise theoretical results facilitate direct applications of topological grid-like frame structures in industry and engineering.