  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.
