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Topologically gapless edge states, characterized by topological invariants and Berry's phases of bulk energy bands, provide amazing techniques to robustly control the reflectionless propagation of electrons, photons and phonons. Recently, a new family of topological phases, dictated by the bulk polarization, has been observed, leading to the discovery of the higher-order topological insulators (HOTIs). So far, the HOTIs are only demonstrated in discrete mechanical and electromagnetic systems and electrical circuits with the quantized quadrupole polarization.Here, we realize the higher-order topological states in a two-dimensional (2D) continuous elastic system whose energy bands can be well described. We experimentally observe the gapped onedimensional (1D) edge states, the trivially gapped zero-dimensional (0D) corner states and the topologically protected 0D corner states. Compared with the trivial corner modes, the topological ones, immunizing against defects, are robustly localized at the obtuse-angled but not the acuteangled corners. The topological shape-dependent corner states open a new route for the design of the topologically-protected but reconfigurable 0D local eigenmodes and provide an excellent platform for the topological transformation of elastic energy among 2D bulk, 1D edge and 0D corner modes.

Topological phononic states, facilitating acoustic unique transports immunizing to defects and disorders, have significantly revolutionized our scientific cognition of acoustic wave systems. Up to now, the theoretical and experimental demonstrations of topologically protected one-way transports with pseudospin states in a phononic crystal beyond the graphene lattice with C 6v symmetry are still unexploited. Furthermore, the tunable topological states, in form of robust reconfigurable acoustic pathways, have been evaded in the topological phononic insulators. Here, we realize a topological phase transition in the double Dirac degenerate cone of rotatable triangular phononic crystals with C 3v symmetry, by introducing the zone folding mechanism. Along a topological domain wall between two portions of phononic crystals with distinct topological phases, we experimentally observe the quantum spin Hall (QSH) effect for xiabz2013@hnu.edu.cn (Baizhan Xia) djyu@hnu.edu.cn (Dejie Yu) liujian@hnu.edu.cn (Jian Liu)

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