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This study investigates the topological behavior of a continuous elasticphononic structure characterized by a 3-term Kekul\'e distortion. The elasticwaveguide consists of a hexagonal unit cell whose geometric dimensions areintentionally perturbed according to a generalized Kekul\'e scheme. Theresulting structure exhibits an effective Hamiltonian that resembles a quantumspin Hall system, hence suggesting that the waveguide can support helicaltopological edge states. Using first-principles calculations, we show theexistence of nondegenerate pseudospin states and a very peculiar 6-lobepseudospin texture and Berry curvature pattern. Important insights are alsoprovided concerning the topological states in the Kekul\'e lattice, so farconsidered indistinguishable, and their critical role in enabling uniquegapless edge states, typically not achievable in phononic systems.

Non-conventional topological band properties and gapless helical edge states in elastic phononic waveguides with Kekul\'e distortion