Material vs. Structure: Topological Origins of Band-Gap Truncation Resonances in Periodic Structures

While resonant modes do not exist within band gaps in infinite periodic materials, they may appear as in-gap localized edge modes once the material is truncated to form a finite periodic structure. Here, we provide an analysis framework that reveals the topological origins of truncation resonances, elucidating formally the conditions that influence their existence and properties. Elastic beams with sinusoidal and step-wise property modulations are considered as classical examples of periodic structures. Their non-trivial topological characteristics stem from the consideration of a phason parameter that produces spatial shifts of the property modulation while continuously varying how the boundaries are truncated. In this context, non-trivial band gaps are characterized by an integer topological invariant, the Chern number, which is equal to the number of truncation resonances that traverse a band gap as the phason is varied. We highlight the existence of multiple chiral edge states that may be localized at opposite boundaries, and illustrate how these can be independently tuned by modified boundary-specific phason parameters. Furthermore, we show that the frequency location of a truncation resonance is influenced by the modulation volume fraction, boundary conditions, and number of cells comprising the finite structure, thus quantifying its robustness to these factors. Non-topological in-gap resonances induced by a defect are also demonstrated, showing that these can be coupled with topological modes when the defect is located at an edge. Finally, experimental investigations on bi-material phononic-crystal beams are conducted to support these findings. The tunability of truncation resonances by material-property modulation may be exploited in applications ranging from vibration attenuation and thermal conductivity reduction to filtering and flow control by phononic subsurfaces.