Band Gap Scalability in Optimised Phononic Crystals

Phononic crystals (PnCs) are distinguished by their exceptional ability to control the propagation of elastic and acoustic waves in a medium, resulting in the attenuation of wave propagation within specific frequency ranges known as band gaps. This property enables promising engineering applications as metamaterials in fields such as seismic engineering, piezoelectric control, sensing, and sound absorption. Although significant efforts have been made to optimise the design of these metamaterials to maximise band gap width, the relationship between band gap location, size, and scaling laws has not been explicitly established. In this work, we investigate the relationship between band gap frequency, width, and structural scaling. We analyse PnCs from the literature with optimised band gaps, incorporating different types of finite elements, such as truss, beam, and two-dimensional elements, to enhance the scalability analysis. The case studies include three unit cell types: truss-like lattices, two-dimensional plates, and sandwich panels. The results demonstrate a consistent inverse proportionality between band gap frequency and length scale across all studied cases, providing a straightforward scalability rule. Additionally, the study highlights that deviations from strict geometric similarity, often required due to manufacturing constraints or geometric limitations, result in predictable yet non-linear variations in relative band gap properties. Understanding these deviations is crucial for realistic design scenarios, enabling designers to leverage pre-optimised structures effectively, reducing computational effort, and supporting practical applications of phononic metamaterials.