A Quantitative Study of Energy Localization Characteristics in Defect-embedded Phononic Crystals

Phononic crystals (PnCs) are periodic engineered media that can customize the spatio-temporal characteristics of mechanical energy propagation. PnCs that additionally leverage precisely embedded defects can achieve robust energy localization with desirable spatio-temporal characteristics, opening avenues for critical engineering applications, e.g., energy harvesting, waveguiding, and fluid flow control. Numerous studies have qualitatively explored the localized dynamics via simulations and experiments, investigating the defect resonance frequency as the primary feature. However, the frequency represents only a subset of the relevant characteristics and a systematic approach to quantify the full scope of the defect dynamics remains elusive. This article establishes the frequency, mode shape, and localized velocity (or displacement) amplitude envelope as three significant factors governing the defect resonance dynamics, and quantitatively examines these characteristics using a modified version of the perturbed tridiagonal n-Toeplitz method. The proposed method accurately estimates the resonance characteristics in 1D and 2D defect-embedded PnC lattices with single and multiple defects and elucidates the effects of damping. The method is used to highlight how the key characteristics of defect modes depend on system parameters. Finally, we demonstrate the benefits of defect modes through two defect-based PnCs that can accommodate -- (i) a virtual ground, and (ii) achieve customized acoustic interaction and absorption, and use the proposed method to analyze these scenarios. The proposed strategy can be readily extended to more elaborate PnCs and augments the design space for defect-based PnCs.