Controlling Mechanical Response in Metamaterials via Topological States of Matter

Abstract

Flexible metamaterials are defined by repeating structural patterns that permit them to possess exciting elastic properties, such as robust, programmable soft deformations. In recent years, they have profited from topological states of matter, in that their mechanical responses are protected against nonlinearities and disorder. These topological states generate a bulk-boundary correspondence, in which the bulk structure guarantees the existence of (mechanically) conducting edge states on the boundary. While recent work by Kane and Lubensky derived these surface effects for mechanical lattices that counted for equal number of degrees of freedom and constraints, we extend the topological paradigm in two ways: lattices that are over-constrained and continuum systems. For the former, we prove that novel deformation modes in rigid metamaterials can be found on lower-dimensional elements of the surface dictated by the degree of coordination of the unit cell. For the latter, we derive a continuum theory which uses the energetic cost of applied strains on a microstructure to derive an elastic bulk-boundary correspondence that is topological in nature. In order to observe topological soft deformations, these systems rely on a microscopic relaxation mechanism that accounts for both short-distance rearrangements of the cell and for long-wavelength strains. Finally, we use the topological nature of well-established metamaterials to quantify the mechanical response of non-ideal systems, such as those hyperstatic in nature or those with weak disorder. This work includes the introduction of a polarization and a geometric phase to quantify boundary effects and bulk response, respectively. Overall, the purpose of this dissertation is to expand the theory of topological modes to a greater variety of flexible metamaterials.