Most previously reported Dirac valley degeneracies in continuous phononic crystal plates originate from Bragg scattering of the structures and generally have only single-band elastic-wave topological edge states. In the present work, a pair of triangular prisms is used in the construction of hexagonal-lattice phononic crystal plates to mimic the dual-band elastic valley Hall effect. Based on the spatial inversion symmetry conditions, which are related to the intrinsic frequencies of the resonators, the valley degeneracies, topological nontrivial bandgaps, and energy band inversion characteristics of multiple resonance modes are investigated by using the finite element method. Edge passbands combining distinct topology phases exist in each of the two nontrivial bandgaps of the ribbon configuration. The full-field simulations for flexural waves in the waveguide structure are demonstrated to support topologically valley-protected edge transmission in both bands, which immunizes the transport against backscattering from large corners and defects in the route. This work provides a reference for valley edge protection in subwavelength continuous elastic plate media and for the manipulation of the elastic waves at multiple frequencies.

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The subwavelength artificial elastic metamaterials can be used for efficient transmission of vibration energies in a tunable frequency range. A local-resonance equal-thickness elastic metamaterial plate is designed in this work. The triangular resonators in the rhombic unit cell are used to exploit the valley degeneracy of the out-of-plane locally resonant modes under hexagonal symmetry conditions. The resonators consist of high-density cores wrapped by soft elastic material, and the differences in the intrinsic frequencies produce a topological band gap caused by the double resonant effect. The reversal of the energy bands proves that the band gap has a non-trivial topology effect, i.e., there exists edge states with robustness and high transmission efficiency. This structure can be utilized in railway transportation as an advanced elastic wave medium material to improve the efficiency of sensing and harvesting the energy of vertical vibrations between vehicles and rails.

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