The topologically protected elastic wave transport in hierarchical metamaterials for the in-plane mode and out-of-plane mode is investigated in this work. First of all, the influence of two key parameters (the hierarchical order and length ratio) on dispersion relation is studied in detail. It can be found that the introduction of structural hierarchy is beneficial for the formation of low-frequency Dirac cones (DCs). But the number of DCs is independent of the hierarchical order. Moreover, the frequency of DCs and band gaps can be tuned effectively by the length ratio. Tuning the lower-order length ratio is more effective to form low-frequency DCs. Based on above findings, we proposed novel topological hierarchical metamaterials analogue to quantum spin Hall effects (QSHEs) by zone-folding method. By shrinking or expanding the supercell, two types of supercells with distinct topological property are built. In addition, through the dispersion analysis and full-field numerical simulation, topological interface states for both wave modes are obtained successfully, and the robustness is verified also. To overcome the limited frequency range of topological interface states, we try to introduce the temperature-sensitive epoxy into the topological hierarchical metamaterial. Fortunately, the frequency of topological interface states can be tuned in a wide range, and the energy concentration cannot be affected by the temperature. This work will broaden the applied range of hierarchical metamaterials, and is expected to meet the tremendous needs for multifunctional materials in engineering practice.

在这项工作中，研究了面内模式和面外模式的分层超材料中的拓扑保护弹性波传输。首先，详细研究了两个关键参数（层次顺序和长度比）对色散关系的影响。可以发现，结构层次的引入有利于低频狄拉克锥（DC）的形成。但是 DC 的数量与等级顺序无关。此外，DC的频率和带隙可以通过长度比进行有效调整。调整低阶长度比可以更有效地形成低频 DC。基于上述发现，我们通过区域折叠方法提出了类似于量子自旋霍尔效应 (QSHE) 的新型拓扑分层超材料。通过缩小或扩大超级晶胞，构建了两种具有不同拓扑特性的超晶胞。此外，通过频散分析和全场数值模拟，成功获得了两种波模的拓扑界面态，并验证了鲁棒性。为了克服拓扑界面态的有限频率范围，我们尝试将温度敏感的环氧树脂引入到拓扑分层超材料中。幸运的是，拓扑界面态的频率可以在很宽的范围内调节，并且能量集中不受温度的影响。该工作将拓宽多级超材料的应用范围，有望满足工程实践中对多功能材料的巨大需求。通过频散分析和全场数值模拟，成功获得了两种波模态的拓扑界面态，并验证了鲁棒性。为了克服拓扑界面态的有限频率范围，我们尝试将温度敏感的环氧树脂引入到拓扑分层超材料中。幸运的是，拓扑界面态的频率可以在很宽的范围内调节，并且能量集中不受温度的影响。该工作将拓宽多级超材料的应用范围，有望满足工程实践中对多功能材料的巨大需求。通过频散分析和全场数值模拟，成功获得了两种波模态的拓扑界面态，并验证了鲁棒性。为了克服拓扑界面态的有限频率范围，我们尝试将温度敏感的环氧树脂引入到拓扑分层超材料中。幸运的是，拓扑界面态的频率可以在很宽的范围内调节，并且能量集中不受温度的影响。该工作将拓宽多级超材料的应用范围，有望满足工程实践中对多功能材料的巨大需求。为了克服拓扑界面态的有限频率范围，我们尝试将温度敏感的环氧树脂引入到拓扑分层超材料中。幸运的是，拓扑界面态的频率可以在很宽的范围内调节，并且能量集中不受温度的影响。该工作将拓宽多级超材料的应用范围，有望满足工程实践中对多功能材料的巨大需求。为了克服拓扑界面态的有限频率范围，我们尝试将温度敏感的环氧树脂引入到拓扑分层超材料中。幸运的是，拓扑界面态的频率可以在很宽的范围内调节，并且能量集中不受温度的影响。该工作将拓宽多级超材料的应用范围，有望满足工程实践中对多功能材料的巨大需求。