Topologically protected waves in classical media provide unique opportunities for one-way wave transport and immunity to defects. Contrary to acoustics and electromagnetics, their observation in elastic solids has so far been elusive because of the presence of multiple modes and their tendency to hybridize at interfaces. Here, we report on the experimental investigation of topologically protected helical edge modes in elastic plates patterned with an array of triangular holes, along with circular holes that produce an accidental degeneracy of two Dirac cones. Such a degeneracy is subsequently lifted by careful breaking of the symmetry along the thickness direction, which emulates the spin orbital coupling in the quantum spin Hall effect. The joining of two plates that are mirror-symmetric copies of each other about the plate midthickness introduces a nontrivial interface that supports helical edge waves. The experimental observation of these topologically protected wave modes in elastic continuous plates opens avenues for the practical realization of structural components with topologically nontrivial waveguiding properties and their application to elastic waveguiding and confinement.

Topological protection offers unprecedented opportunities for wave manipulation and energy transport in various fields of physics, including elasticity, acoustics, quantum mechanics, and electromagnetism. Distinct classes of topological waves have been investigated by establishing analogs with the quantum, spin, and valley Hall effects. We here propose and experimentally demonstrate the possibility of supporting multiple classes of topological modes within a single platform. Starting from a patterned elastic plate featuring a double Dirac cone, we create distinct topological interfaces by lifting such degeneracy through selective breaking of symmetries across the thickness and in the plane of the plate. We observe the propagation of a new class of heterogeneous helical-valley edge waves capable of isolating modes on the basis of their distinct polarization, i.e., the specific mode wave field distribution within the unit cell. Our results show the onset of wave splitting resulting from the interaction of multiple topological equal-frequency wave modes, which may have significance in applications involving elastic beam splitters, switches, and filters.

The study of vibrational properties in engineered periodic structures relies on the early intuitions of Haüy and Boscovich, who regarded crystals as ensembles of periodically arranged point masses interacting via attractive and repulsive forces. Contrary to electromagnetism, where mechanical properties do not couple to the wave propagation mechanism, in elasticity this paradigm inevitably leads to low stiffness and high-density materials. Recent works transcend the Haüy-Boscovich perception, proposing shaped atoms with finite size, which relaxes the link between their mass and inertia, to achieve unusual dynamic behavior at lower frequencies, leaving the stiffness unaltered. Here, we introduce the concept of tacticity in spin-spin-coupled chiral phononic crystals. This additional layer of architecture has a remarkable effect on their dispersive behavior and allows to successfully realize material variants with equal mass density and stiffness but radically different dynamic properties.

The metamaterial paradigm has allowed an unprecedented space-time control of various physical fields, including elastic and acoustic waves. Despite the wide variety of metamaterial configurations proposed so far, most of the existing solutions display a frequency response that cannot be tuned, once the structures are fabricated. Few exceptions include systems controlled by electric or magnetic fields, temperature, radio waves and mechanical stimuli, which may often be unpractical for real-world implementations. To overcome this limitation, we introduce here a polymeric 3D-printed elastic metamaterial whose transmission spectrum can be deterministically tuned by a light field. We demonstrate the reversible doubling of the width of an existing frequency band gap upon selective laser illumination. This feature is exploited to provide an elastic-switch functionality with a one-minute lag time, over one hundred cycles. In perspective, light-responsive components can bring substantial improvements to active devices for elastic wave control, such as beam-splitters, switches and filters.

Topological physics is emerging as an active area of research, addressing fundamental questions on how geometry, symmetry, and topology affect physical properties, paving the way toward novel technological applications. Originally investigated in quantum systems, these concepts have been thereafter translated across diverse domains including, electromagnetic, plasmonic, elastic, and acoustic waves. Specifically, in elasticity, due to the strong tendency to hybridize of wave modes with different polarization, topological protection is viewed as a revolutionizing approach to design waveguides supporting unique features such as (i) being immune to defects and (ii) suppressing backscattering during the wave propagation phenomenon. These novel features arise as a consequence of their dispersion surface topology. This Tutorial aims to introduce the theoretical, numerical, and experimental frameworks to investigate topological elastic waveguides, discussing the key ideas, first, in the context of discrete systems, and then, in continuous elastic solids. After a comprehensive description of the currently used state of the art scientific techniques, various classes of topological wave phenomena leading to localized waves in elastic architected plates and beams are presented. Implications of the presence of both longitudinal and shear waves in elastic solids are discussed, and the associated challenges, opportunities, and strategies to exploit their interplay highlighted. The symmetry conditions required to induce them are discussed using a number of representative examples. Finally, future research directions of this fledgling field are outlined.