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Phononic crystals represent a class of architected materials with spatially periodic unit cells. In their most common form, they involve distinct layers with sharply contrasting properties (e.g., hard-soft materials). These systems exhibit wide frequency bandgaps which are understood to culminate from impedance mismatches between the neighboring layers. This work focuses on a less-commonly studied configuration of phononic crystals in which the unit cell is formed via a continuous harmonic variation of material properties. While multi-layered solid phononic crystals display an infinite number of bandgaps, we show that an elastically-graded phononic crystal exhibits a single, well-defined bandgap that can be tuned by varying the parameters of the harmonically-varying stiffness. Additionally, we reveal that such graded phononic crystals transition to a fully non-dispersive behavior above a frequency threshold, rendering them pseudo-homogeneous structures despite their physical heterogeneity. As the unit cell transforms from distinct layers to a smoothly graded profile, we seek to understand the system’s evolution leading to the closing of higher frequency bandgaps and the loss of dispersive wave response. We employ a segmented discretization of the harmonic profile to approximate its continuous form with the goal of deriving a semi-analytical solution that systematically captures the emergence, width, and eventual closure of the bandgaps. By varying the resolution of discretization, we gain valuable insight into the changes in bandgap widths and the subsequent morphing of the dispersion bands, which enables the design of broadband phononic crystals with tailored and reconfigurable wave manipulation capabilities.
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The Role of Frequency and Impedance Contrasts in Bandgap Closing and Formation Patterns of Axially-Vibrating Phononic Crystals
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 versus even bandgaps) or location (low versus 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.
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- Award ID(s):
- 1847254
- PAR ID:
- 10473234
- Publisher / Repository:
- ASME
- Date Published:
- Journal Name:
- Journal of Applied Mechanics
- Volume:
- 91
- Issue:
- 3
- ISSN:
- 0021-8936
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1115/1.4063815
