Abstract

Phononic crystals (PnCs) have received growing attention in recent years, due to their ability to manipulate elastic waves, such as in the case of defect-mode-enabled energy localization. Although previous studies have explored defect modes of PnCs – from phenomenon observations to their potential applications – little effort has been made to date to reveal fundamental mechanisms of defect-mode-enabled energy localization. Thus, this study proposes a lumped-parameter analytical model to reveal the underlying principles of the formation of defect bands of a one-dimensional PnC when a single defect is introduced, or the splitting of defect bands when double defects are introduced. Through the investigation of 1) evanescent wave characteristics in the defect-mode shapes, and 2) the asymptotically equivalent behaviors of defect bands and defect-mode shapes with limiting behavior approaches, this study demonstrates a new aspect of why a band gap should be the prerequisite for achieving defect-mode-enabled energy localization. It is confirmed that defect-mode shapes are normal modes, rather than propagating wave modes. The key findings of this study are as follows: 1) the exponentially attenuating characteristics of evanescent waves in a band gap generate a fixed-like boundary condition, which surrounds single or double defects, and 2) mechanical resonance, attributed to the fixed-like boundary condition, leads to the formation and splitting of defect bands.