Topologically protected edge states of elastic waves in phononic crystal plates
Authors: Zheng Zhou-Fu , Yin Jian-Fei , Wen Ji-Hong , Yu Dian-Long ,   Laboratory of Science and Technology on Integrated Logistics Support, College of Intelligence Science and Technology, National University of Defense Technology, Changsha 410073, China
Published: December 2020

<sec> The topologically protected edge states of elastic waves in phononic crystal plates have the outstanding characteristics in wave manipulation such as the strong suppression of back-scattering and defect immunity, which can be used for controlling vibration and noise, detecting the structural damage, conducting the material nondestructive test and other engineering practices, and therefore have received much attention. But for plate structures, the propagation of elastic waves is complicated due to the coexistence and coupling of different types of wave modes, resulting in a challenge in designing topologically protected states. </sec><sec>In this paper, a simple phononic crystal plate with triangular holes is designed for elastic wave manipulation based on topologically protected edge states. The band structure characteristics of the unit cell are studied by varying the rotation angle <i>θ</i> of the triangular holes around their geometric centers from the initial positions. It is found that the band structure of the initial unit cell with rotation angle <i>θ</i> = 0° has two pairs of degenerate modes. At <inline-formula><tex-math id="M6">\begin{document}$ \theta = \pm 33^\circ $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200542_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200542_M6.png"/></alternatives></inline-formula>, a double Dirac cone appears at the center <i>Γ</i> point of the Brillouin zone without requiring the lattices to fold, and a band inversion occurs on both sides of <inline-formula><tex-math id="M7">\begin{document}$ \pm 33^\circ $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200542_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="15-20200542_M7.png"/></alternatives></inline-formula> which can be characterized as a topological phase transition. </sec><sec>The elastic band gap and two kinds of pseudospin states with clockwise or counterclockwise circulating mechanical energy flux patterns in the band structure are found by calculating the projected band structures of a supercell which is composed of phononic crystals with different topological phases. Based on this finding, different constructions of phononic waveguide are used for implementing the numerical analysis to demonstrate the back-scattering immunity of the edge states when disorder, tortuosity and cavity are introduced into the waveguide. Unidirectional robust propagation and multichannel waveguide switch due to the pseudospin-dependent one-way edge modes are also validated with numerical models. The phononic crystal plate presented in this paper provides a simple realizable method of designing the topologically protected elastic edge states. </sec>