The study of phononic materials and structures is an emerging discipline that lies at the crossroads of vibration and acoustics engineering and condensed matter physics. Broadly speaking, a phononic medium is a material or structural system that usually exhibits some form of periodicity, which can be in the constituent material phases, or the internal geometry, or even the boundary conditions. As such, its overall dynamical characteristics are compactly described by a frequency band structure, in analogy to an electronic band diagram. With roots extended to early studies of periodic systems by Newton and Rayleigh, the field has grown to encompass engineering configurations ranging from trusses and ribbed shells to phononic crystals and metamaterials. While applied research in this area has been abundant in recent years, treatment from a fundamental mechanics perspective, and particularly from the standpoint of dynamical systems, is needed to advance the field in new directions. For example, techniques already developed for the incorporation of damping and nonlinearities have recently been applied to wave propagation in phononic materials and structures. Similarly, numerical and experimental approaches originally developed for the characterization of conventional materials and structures are now being employed toward better understanding and exploitation of phononic systems. This article starts with an overview of historical developments and follows with an in-depth literature and technical review of recent progress in the field with special consideration given to aspects pertaining to the fundamentals of dynamics, vibrations, and acoustics. Finally, an outlook is projected onto the future on the basis of the current trajectories of the field.

We report on the experimental observation of topologically protected edge waves in a two-dimensional elastic hexagonal lattice. The lattice is designed to feature $K$-point Dirac cones that are well separated from the other numerous elastic wave modes characterizing this continuous structure. We exploit the arrangement of localized masses at the nodes to break mirror symmetry at the unit-cell level, which opens a frequency band gap. This produces a nontrivial band structure that supports topologically protected edge states along the interface between two realizations of the lattice obtained through mirror symmetry. Detailed numerical models support the investigations of the occurrence of the edge states, while their existence is verified through full-field experimental measurements. The test results show the confinement of the topologically protected edge states along predefined interfaces and illustrate the lack of significant backscattering at sharp corners. Experiments conducted on a trivial waveguide in an otherwise uniformly periodic lattice reveal the inability of a perturbation to propagate and its sensitivity to backscattering, which suggests the superior waveguiding performance of the class of nontrivial interfaces investigated herein.

One of the outstanding challenges in phononic crystals and acoustic metamaterials development is the ability to tune their performance without requiring structural modifications. We report on the experimental demonstration of a tunable acoustic waveguide implemented within a two-dimensional phononic plate. The waveguide is equipped with a periodic array of piezoelectric transducers which are shunted through passive inductive circuits. The resonance characteristics of the shunts lead to strong attenuation and to negative group velocities at frequencies defined by the circuits' inductance. The proposed waveguide illustrates the concept of a controllable acoustic logic port or of an acoustic metamaterial with tunable dispersion characteristics.

Elastic/acoustic metamaterials made from locally resonant arrays can exhibit bandgaps at wavelengths much longer than the lattice size for various applications spanning from low-frequency vibration/sound attenuation to wave guiding and filtering in mechanical and electromechanical devices. For an effective use of such locally resonant metamaterial concepts in finite structures, it is required to bridge the gap between the lattice dispersion characteristics and modal behavior of the host structure with its resonators. To this end, we develop a novel argument for bandgap formation in finite-length elastic metamaterial beams, relying on the modal analysis and the assumption of infinitely many resonators. We show that the dual problem to wave propagation through an infinite periodic beam is the modal analysis of a finite beam with an infinite number of resonators. A simple formula that depends only on the resonator natural frequency and total mass ratio is derived for placing the bandgap in a desired frequency range, yielding an analytical insight and a rule of thumb for design purposes. A method for understanding the importance of a resonator location and mass is discussed in the context of a Riemann sum approximation of an integral, and a method for determining the optimal number of resonators for a given set of boundary conditions and target frequency is introduced. The simulations of the theoretical framework are validated by experiments for bending vibrations of a locally resonant cantilever beam.