[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] Topological insulators are new phases of matter whose properties are derived from a number of qualitative yet robust topological invariants rather than specific geometric features or constitutive parameters. Their salient feature is that they conduct localized waves along edges and interfaces with negligible scattering and losses induced by the presence of specific varieties of defects compatible with their topological class. Following the explosion of activities in the structure of topological phases in quantum mechanics and condensed matter physics, mechanical and acoustic waves have recently emerged as excellent platforms that exemplify the universality and diversity of topological phases. This dissertation is part of this development to help bridge a gap between quantum mechanical constructs and their potential applications in classical mechanics and acoustics. First, we respectively implement mechanical analogues to quantum valley and quantum spin Hall effects in the mass-spring Kagome lattices. Therein, our main tool is asymptotic homogenization technique that transforms the discrete motion equation of the lattice into a continuum partial differential equation. Throughout the study, topological Stoneley waves localized at the interface between two Kagome lattices are fully characterized in terms of existence conditions, modal shapes, decay rates and group velocities. Both quantum valley and quantum spin Hall insulators inducing perturbations must preserve time-reversal symmetry. By deliberately breaking this symmetry, we investigate Thouless pumping and the quantum Hall effect in 1D and 2D periodically time-modulated elastic materials, respectively. We theoretically and numerically demonstrate the existence of topologically protected one-way edge states immune to scattering by sharp corners, defects, randomly disordered modulation phases, and dissipation effects. However, the physical realization of a temporally controlled quantum Hall effect that produces resilient transport has been proved to be extremely difficult. To address this obstacle, we then utilize pumping parameters in space as synthetic dimensions instead of time to realize higher-dimensional topological models in platforms with lower dimensionality. By adiabatically transforming pumping parameters along the synthetic dimension, we observe topologically protected sound propagation, which is smoothly pumped from one side to the other. Furthermore, we introduce a new class of topological insulators characterized by nontrivial bulk polarization. In addition to well-studied topological edge states, these systems have been shown to host Wannier-type second-order corner states. By manipulating simply lattice geometry and quantized characterization, we investigate numerically and demonstrate experimentally the topologically protected in-plane edge and corner states in a mechanical Kagome lattice. The topological protected states can either be at finite frequencies or at zero frequency. As such, we finally demonstrate topological zero-frequency deformation modes in a fairly generic isostatic truss: Kagome lattices. We propose a new effective medium theory baptized "microtwist elasticity" capable of rendering topological polarization effects on a macroscopic scale. Various numerical and analytical calculations, of the shape and distribution of zero modes, of dispersion diagrams and of polarization effects, systematically show the quality of the proposed effective medium theory.
Topological gaps by twisting
