The role of frequency and impedance contrasts in bandgap closing and formation patterns of axially-vibrating phononic crystals

TL;DR

This paper develops a transfer-matrix-based framework to analyze how frequency and impedance contrasts influence bandgap formation, closing, and patterns in one-dimensional phononic crystals, enhancing understanding beyond traditional Bloch-wave predictions.

Contribution

It introduces a novel analytical approach connecting physical origins of bandgap behaviors to contrast parameters, applicable to multi-layered PnCs for tailored bandgap design.

Findings

Characterizes bandgap patterns using dimensionless contrast constants.

Connects bandgap closing to Bragg condition and natural resonances.

Generalizes framework for any number of layers in PnCs.

Abstract

Bandgaps, or frequency ranges of forbidden wave propagation, are a hallmark of Phononic Crystals (PnCs). Unlike their lattice counterparts, PnCs taking the form of continuous structures exhibit an infinite number of bandgaps of varying location, bandwidth, and distribution along the frequency spectrum. While these bandgaps are commonly predicted from benchmark tools such as the Bloch-wave theory, the conditions that dictate the patterns associated with bandgap symmetry, attenuation, or even closing in multi-bandgap PnCs remain an enigma. In this work, we establish these patterns in one-dimensional rods undergoing longitudinal motion via a canonical transfer-matrix-based approach. In doing so, we connect the conditions governing bandgap formation and closing to their physical origins in the context of the Bragg condition (for infinite media) and natural resonances (for finite counterparts). The developed framework uniquely characterizes individual bandgaps within a larger dispersion spectrum regardless of their parity (i.e., odd vs even bandgaps) or location (low vs high-frequency), by exploiting dimensionless constants of the PnC unit cell which quantify the different contrasts between its constitutive layers. These developments are detailed for a bi-layered PnC and then generalized for a PnC of any number of layers by increasing the model complexity. We envision this mathematical development to be a future standard for the realization of hierarchically-structured PnCs with prescribed and finely tailored bandgap profiles.