Theoretical and Experimental Analysis of Topological Elastic Waveguides

Ting-Wei Liu, Purdue University

Abstract

The capability of manipulation of the flow of mechanical energy in the form of mechanical waves (including acoustic and elastic waves) has always been a challenge and a critical part in various areas of engineering. The recent advances in topological acoustic/elastic metamaterials certainly open a new pathway to the manipulation of mechanical waves, especially for the novel scattering-immune wave-guiding capability, even in the presence of defects, disorders or sharp bends along the waveguide. In this Dissertation, the theoretical background and experimental evidence of various types of elastic-wave topological metamaterials including analogues to 2D quantum valley Hall effect (QVHE) materials, 2D quantum spin Hall effect (QSHE) topological insulators are presented. First, the formulation the elastic-wave analogue to QVHE materials in a general continuous elastic phononic structure (not limited to local resonant lattices, filling the gap in the literature) is proposed, and a strategy using pressurized cells to actively control the phononic lattice is presented. By finite prestrain and geometric nonlinear effect, the space inversion symmetry of the original hexagonal lattice is broken, resulting in distinct QVHE phases (characterized by valley Chern numbers) in lattice domains with opposite pressurization. With such mechanism, the edge-state path, i.e., the domain wall connecting lattices with distinct QVHE phases, can be real-time configured. Further more, edge states with tunable ω-k dispersion can be created at the external boundaries of the lattice by appropriate pressurization of the outermost cells. An aluminum reticular sheet built with water-jet cutting is machined in the pre-deformed pattern with a Z-shape domain wall at the center, which spatially divides the sheet into two domains with opposite QVHE phases. Using piezoelectric transducers and laser Doppler vibrometry, the measured harmonic and transient responses confirm the back-scattering-immunity of the topological edge states, and the ω-k dispersion matches the numerical prediction. A strategy is proposed for unidirectionally generating edge states along the domain wall using two offphase transducers, which is also experimentally demonstrated. For elastic-wave analogue to QSHE topological insulators, we focus on the “zone-folding” method and propose a honeycomb 2D elastic beam network with periodically altered thickness with a generalized Kekulé distortion pattern. Such framework provides a parametric space with exhaustive control in the topological phase diagram of waves in the lattice compared to earlier works in the literature. The effective Hamiltonian as well as the characterized topological phase are gauge dependent, particularly they change with different reference frames. This lead to ambiguity in the topological phase of such phononic crystal. Based on this argument, it is predicted that edge states could exist at a dislocation interface connecting two piece of phononic structures of the same pattern with relative displacement. Following the same idea, but considering the available fabrication options, a phononic plate with honeycomb groove pattern engraved on both sides is built, which the depth varied according to the Kekulé pattern.With proper tuning of the parameters, it realizes an analogue to the QSHE topological insulator.With ab initio calculation of the Berry curvature (without involving any approximations such as the perturbative approach), a new topological invariant local topological charge is defined and evaluated as the counterpart of the Z2invariant in the classical-wave-zone-folding analogue.

Degree

Ph.D.

Advisors

Semperlotti, Purdue University.

Subject Area

Acoustics|Electromagnetics|Physics

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