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In solid state physics, a bandgap (BG) refers to a range of energies where no electronic states can exist. This concept was extended to classical waves, spawning the entire fields of photonic and phononic crystals where BGs are frequency (or wavelength) intervals where wave propagation is prohibited. For elastic waves, BGs are found in periodically alternating mechanical properties (i.e., stiffness and density). This gives birth to phononic crystals and later elastic metamaterials that have enabled unprecedented functionalities for a wide range of applications. Planar metamaterials are built for vibration shielding, while a myriad of works focus on integrating phononic crystals in microsystems for filtering, waveguiding, and dynamical strain energy confinement in optomechanical systems. Furthermore, the past decade has witnessed the rise of topological insulators, which leads to the creation of elastodynamic analogs of topological insulators for robust manipulation of mechanical waves. Meanwhile, additive manufacturing has enabled the realization of 3D architected elastic metamaterials, which extends their functionalities. This review aims to comprehensively delineate the rich physical background and the state‐of‐the art in elastic metamaterials and phononic crystals that possess engineered BGs for different functionalities and applications, and to provide a roadmap for future directions of these manmade materials.

 

Recent discoveries in twisted heterostructure materials have opened research directions in classical wave systems. This Letter investigates a family of double-sided pillared phononic crystal plates as the elastodynamic analog of bilayer graphene, including twisted bilayer graphene. The phononic crystal plate design is first validated by studying the basic AA- and AB-stack configurations under weak interlayer coupling. A specific commensurate twist angle giving rise to the sublattice exchange even symmetry is then studied to examine the twist-modulated band structure. Finally, this study demonstrates that the same twist angle, in concert with an ultra-strong interlayer coupling, can collectively create valley-dependent edge states that have not been previously observed in electronic bilayer graphene.