On the non-dispersive and pseudo-homogeneous behavior of harmonically-graded phononic crystals at high frequency regimes

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.