Topology in Linear Mechanical Metamaterials

This thesis deals with topological effects in linear mechanical metamaterials governed by Newton's equation of motion. Mechanical metamaterials are artificially designed structures whose behavior arises from a scale much larger than its constituting units. The topological effects taken into account are the ones known from single-particle condensed matter physics. While first introduced in the context of quantum mechanics, they are not bound to it and appear in other contexts as well. As such, the field of topological mechanical metamaterials is the youngest offspring implementing ideas from topological band theory and beyond. This thesis is part of this development and its contributions are fourfold. The first contribution is a systematic approach to import topological effects from single-particle condensed matter physics to mechanical metamaterials. We show how to bring Newton's equation of motion into a form akin the Schrödinger equation. This then allows for a direct import of the desired physics. Besides, through this transformation we combine previously unconnected approaches to carry over topological effects, and set out how further ones transfer. A central feature of topological materials is the presence of robust surface modes that are of interest in sight of applications. However, some types of topological effects can only be implemented if classical time-reversal symmetry is broken, which in turn can be hard to achieve for versatile and affordable mechanical metamaterial. We theoretically and experimentally demonstrate how a passive, time-reversal-invariant topological material can be built, by implementing the topological aspects of the quantum spin Hall effect in a mechanical metamaterial. We experimentally characterize the edge (surface) modes and discuss their topological protection. The quantum spin Hall effect is protected by quantum time-reversal symmetry, which translates into a local symmetry in its mechanical version. By deliberately breaking this symmetry, we show how the topological edge channels can be switched on and off. Furthermore, we experimentally show that it is sufficient to break the symmetry in a very small spatial region of the system to obtain an almost perfect switching behavior. Taking into account additional energy terms renders topological invariants often ill-defined. Nevertheless, they offer a structured approach to engineer peculiar surface physics. We create such a metamaterial with a polar elastic response. When straining the material with a point-like object on a certain surface, it provides a very soft response, whereas when poked on the opposite surface the material is hard. The effect is robust, and in particular not prone to wearing