An overview of geometric phases in elastic systems and their connection to topological invariants of elastic metamaterials

The geometric phase is an additional phase factor acquired by oscillating dynamical systems. It has emerged as an insightful parameter to understand the dynamic behavior in a variety of systems, from molecular physics to elastic waveguides. In more recent years, the geometric phase has been widely exploited in connections with the analysis of topological materials. The present article reviews the concept of geometric phase in elastic systems and its connection to the design of elastic topological metamaterials. Examples are presented to explain the theoretical basis of the geometric phase by using arguments of differential geometry and topology. These concepts are then applied to the analysis of a one-dimensional elastic topological metamaterial that possesses localized vibration modes immune to geometric perturbations.