Amplitude-dependent boundary modes in topological mechanical lattices

Abstract

Abstract Boundary modes localized at the edge or on the interface of topological mechanical lattices are analogous to their electronic counterparts in topological insulators and are robust and immune to structural imperfections. Most studies on the boundary modes in mechanical lattices are based on band theories, focusing on amplitude-independent behaviors at different frequency ranges. However, highly distorted topological mechanical lattices exhibit amplitude-dependent nonlinear effects that are difficult to characterize. In this study, a topological mechanical lattice exhibiting amplitude-dependent boundary modes is explored to uncover the evolution and connection between boundary modes in the linear and nonlinear regimes. Both discrete and continuum models are developed. An iterative method for the discrete model is proposed to numerically capture the boundary mode evolution while, based upon the continuum model, an analytical solution of the decay length scale of the boundary modes is obtained and can reduce to the linear band theory prediction at the small amplitude limit. A design strategy is presented for programmable topological polarization in the lattice that can be used to encode mechanical information. The obtained results provide insight into the modulation of topological protected mechanical response and stimulation of spatially ordered modes.