In recent years, the advances in topological insulator in the fields of condensed matter have been extended to classical wave systems such as acoustic and elastic waves. However, the quantitative robustness study of topological states which is indispensable in practical realization is rarely reported. In this work, we proposed topologically protected edge states with zigzag, bridge and armchair interfaces in a new twisted phononic plate. The robustness of non-trivial band gap in bulk structure is clearly presented versus twisted angles, revealing a threshold of 5 degrees which is the key fundamental information for the robustness of topological edge states. We further defined a localized displacement ratio as an efficient parameter to characterize edge states. Due to the different orientation of the three interfaces, zigzag and bridge edge states show higher quantitative robustness in their localized displacement ratio. A map of robustness as a function of both frequency and twisted angle highlights the better performance of the topological zigzag edge state. Robustness is evaluated for twisted angle and for all possible types of interfaces for the first time, which benefits for the design and fabrication of solid functional devices with great potential applications.

近年来，拓扑绝缘体在凝结物领域中的进展已扩展到经典波系统，例如声波和弹性波。然而，很少有关于拓扑状态的定量鲁棒性研究在实际实现中不可缺少的研究。在这项工作中，我们在新的扭曲声子板中提出了带有锯齿形，桥形和扶手椅形界面的拓扑保护边缘状态。相对于扭转角，清楚地展示了块结构中非平整带隙的稳健性，揭示了5度阈值，这是拓扑边缘状态稳健性的关键基础信息。我们进一步定义了局部位移比作为表征边缘状态的有效参数。由于三个界面的方向不同，之字形和桥梁边缘状态在其局部位移比中显示出更高的定量鲁棒性。鲁棒性与频率和扭曲角的关系图突出了拓扑之字形边缘状态的更好性能。首次评估了扭转角和所有可能类型的接口的鲁棒性，这对于具有巨大潜在应用的固体功能器件的设计和制造是有好处的。