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多稳态力学超材料带隙特性及调控研究*

  • 刘宏1,2

    机 构：

    1. 西北工业大学 力学与土木建筑学院,西安 710072

    2. 西北工业大学 复杂系统动力学与控制工业和信息化部重点实验室,西安 710072

    ×
  • ，余江1,2

    机 构：

    1. 西北工业大学 力学与土木建筑学院,西安 710072

    2. 西北工业大学 复杂系统动力学与控制工业和信息化部重点实验室,西安 710072

    ×
  • ，张凯1,2,3

    机 构：

    1. 西北工业大学 力学与土木建筑学院,西安 710072

    2. 西北工业大学 复杂系统动力学与控制工业和信息化部重点实验室,西安 710072

    3. 大连理工大学 工业装备结构分析国家重点实验室,大连 116024

    ×
  • ，邓子辰1,2

    机 构：

    1. 西北工业大学 力学与土木建筑学院,西安 710072

    2. 西北工业大学 复杂系统动力学与控制工业和信息化部重点实验室,西安 710072

    ×

1. 西北工业大学 力学与土木建筑学院,西安 710072；
2. 西北工业大学 复杂系统动力学与控制工业和信息化部重点实验室,西安 710072；
3. 大连理工大学 工业装备结构分析国家重点实验室,大连 116024；

Multi-Stable Mechanical Mematerials for Band Gap Tuning*

  • Liu Hong1,2

    Affiliation：

    1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072 , China

    2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072 , China

    ×
  • ，Yu Jiang1,2

    Affiliation：

    1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072 , China

    2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072 , China

    ×
  • ，Zhang Kai1,2,3

    Affiliation：

    1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072 , China

    2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072 , China

    3. The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024 , China

    ×
  • ，Deng Zicheng1,2

    Affiliation：

    1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072 , China

    2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072 , China

    ×

1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072 , China；
2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an 710072 , China；
3. The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024 , China；

通讯作者:

张凯,E-mail:kzhang@nwpu.edu.cn

中图分类号:O327

文献标识码:A

文章编号:1672-6553-2023-21(7)-005-007

DOI:10.6052/1672-6553-2023-030

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目录contents

    摘要

    多稳态力学超材料具有多重稳定状态和几何重构特性,在变形吸能和能带调控等方面有重要的研究价值.本研究提出了一种多稳态力学超材料.该超材料是由具有双稳态特性的曲杆和相关实体支撑结构组成.结合有限单元法和布洛赫定理,计算了结构的能带结构,分析了不同稳态条件下结构的带隙特性,得到了几何参数对结构带隙的影响规律,并研究了由5×5个单胞组成的阵列结构中部分单胞变形时,整体结构的传输率.研究发现,双稳态连杆的变形状态可显著改变结构的带隙,尤其是在低频范围内产生了更宽的带隙.整体结构中局部一行或一列单胞变形即可在该方向对应带隙频率范围内实现弹性波传播的衰减.

    Abstract

    The multi-stable mechanical metamaterials have the multiple stable status and geometric reconstruction characteristics, which are of great research value in deformation energy absorption and tuning of the band gap. In this study, a multi-stable mechanical metamaterial is proposed,which is composed of bistable curved rods and related solid support structures. Combining the finite element method and Bloch’s theorem, the band structure of the structure is calculated, the band gap characteristics of the structure with different steady-states are analyzed, the effects of geometric parameters on the band gap are obtained, and the transmissions of the array composed of 5×5 unit cells are studied when part of the cells are deformed. It is found that the deformation status of the bistable connecting rod can significantly change the band gap of the structure, especially in the low-frequency range. The elastic wave propagation can be attenuated within the corresponding band gap frequency range by deformation of a row of cells in the structure.

  • 引言

  • 超材料一般指一类具有特殊性质的人造材料，近年来在光学、声学、热学等领域得到了广泛的关注.力学超材料因其优异的静力学和动力学特性，在低频隔振、减振降噪、定向波导、地震波衰减等方面得到了重点的研究[1-7].与传统材料相比，力学超材料特殊的构型，使其具有特定频率的带隙特性，从而实现低频宽带的弹性波传播的阻隔，这为力学超材料在隔振和波动能量吸收等提供了可能.

  • 由于力学超材料的材料选择和结构形式在设计和制造时已经确定，这使得其本身的动力学特性和带隙范围难以有效调节，限制了力学超材料的进一步应用.为了克服此难题，学者们提出了可调谐力学超材料，以更加灵活地调控弹性波在结构中的传播.一方面，通过施加、改变外部条件来调整结构的刚度和质量，如施加电场、磁场、热场等[8，9]，从而改变结构的带隙位置；另一方面，利用具有大变形能力的材料或结构，通过实施结构的大变形来调整带隙，如利用形状记忆效应、屈曲不稳定性等[10-12].

  • 近年来，多稳态力学超结构具有大变形、多稳定状态等优势，逐渐得到了学术界的关注.通过在外部施加激励或载荷，结构变形后可保持在另一个稳定位置.由于其优异的变形能力和吸能特性，被广泛应用于冲击防护、能量吸收、可展开结构设计等[13-18].与此同时，学者们利用此类结构优异的性能，将其应用于结构动力学特性调控和弹性波传播调控的研究中.例如，通过由双稳态连杆组成的多稳态链，可在大振幅激励下，实现结构中过渡波的单向传播和远距离传播[19-22].通过设计一维双稳态单元链构成的可重构超材料，可实现小振幅波传播时良好的可调谐性[23].通过将线性弹簧和非线性双稳弹簧交替布置，得到具有方向可调谐性的超材料，改变波的传播方向[24].另外，基于形状记忆合金设计多稳态力学超材料，可通过外部刺激（荷载或温度场）实现带隙的实时调节，从而进一步优化超材料的振动控制能力[25-27].目前，关于多稳态力学超材料的研究主要集中于静力学性能和变形状态的研究，其动力学性能与变形状态之间的关系和机理尚有待深入研究.

  • 本文提出了一种多稳态力学超材料，结合有限单元法和布洛赫定理，计算结构的能带结构，分析不同稳态条件下的带隙特性，研究几何参数对结构带隙的影响规律.并通过有限元仿真计算结构不同变形情况下的传输率，进一步研究其对振动的衰减效果.

  • 1 多稳态力学超材料动力学模型

  • 1.1 几何模型

  • 本文所研究的力学超材料单胞如图1（a）所示，其可由图1（b）所示的四分之一结构旋转得到.若将该单胞沿x、y方向阵列，则可得到无限大周期晶格结构.该单胞主要由双稳态连杆和实体结构组成.图1（b）为四分之一单元的尺寸，其具体数值列在表1中.材料假设为热塑性聚氨酯弹性体（Thermoplastic Urethane，TPU），其弹性模量、泊松比、密度分别为25MPa，0.25，1280kg/m3.

  • 图1 多稳态力学超材料单胞的几何模型

  • Fig.1 Geometric model of multi-stable mechanical metamaterial unit cell

  • 表1 图1所示单胞的几何参数

  • Table1 The geometric parameters of the unit cell in Fig.1

  • 1.2 能带理论

  • 自由边界单胞的动力学方程为[28]：

  • K-ω2Mq=0
    (1)

  • 其中K和M分别是单胞的整体刚度矩阵和整体质量矩阵，q为单胞位移向量，ω为频率.

  • 根据布洛赫定理，周期晶格单胞中的任意节点位移可以通过相应单胞的节点位移描述，即：

  • qrjn1,n2=qrj0,0en1k1+n2k2
    (2)

  • 式中qrjn1，n2是（n1，n2）单元的位移向量，qrj0，0是参考单元（0，0）的位移向量，（k1，k2）为波数.将式（2）代入式（1），即可得到应用布洛赫定理的周期晶格的动力学方程：

  • Kk1,k2-ω2Mk1,k2qg=0
    (3)

  • 式中K（k1，k2）和M（k1，k2）是考虑周期性边界条件的刚度和质量矩阵.通过沿第一布里渊区不可约部分的边界Γ-Y-X-Γ扫掠波矢，求解式（3）中的特征值，则可得到晶格的能带结构.

  • 2 多稳态力学超材料的静力学性能及能带结构

  • 2.1 多稳态力学超材料的静力学性能

  • 本文的双稳态连杆为曲线形连杆，其几何构型的表达式为[29]：

  • w(x)=h21-cos⁡πxL,x∈[0,L]
    (4)

  • 将四分之一结构两侧固定，在顶端施加指定的位移Δd，并计算顶端反作用力，即可得到力-位移曲线，进而求得双稳态结构第二稳态的位置.当双稳态连杆的跨度L和高度h固定时，厚度t成为影响双稳态行为的最主要因素.图2展示了不同厚度t下的力-位移曲线.当反力为负值并随着位移的增大恢复为零时，结构表现出第二稳态状态，即图2中的B区域.可以看到，随着t的增大，结构达到第二稳定状态的位移值逐渐变小.

  • 图2 四分之一单胞的力-位移曲线

  • Fig.2 Force-displacement curve of the quarter unit cell

  • 2.2 能带结构分析

  • 使用有限元软件分别在x方向和y方向设置Floquet周期边界条件，求解式（3），则可得到结构的能带结构.根据图2计算所得的第二稳态位移值（B区域），通过稳态步骤给结构施加相应位移，根据上述理论可计算得到变形后结构的能带结构.

  • 图3 不同结构变形形式及对应能带图

  • Fig.3 Different structural deformation forms and corresponding band gaps

  • 首先分析t为1mm，a2为10mm时结构的能带图.图3为不同结构变形形式及对应结构前十阶频率的能带图.本文仅研究未变形结构[图3（a）]、单方向变形结构[图3（c）]和双方向变形结构[图3（e）]，其中变形结构是分别在单方向或双方向的双稳态结构施加19.3mm的位移，使其分别达到第二稳态，对应图2中“t=1”曲线反力为零（B区域内）的状态.从图中可以看出，未变形结构在低频范围内有两条带隙，最低带隙在第三阶频率和第四阶频率之间，其频率范围在51.62Hz~84.41Hz，另外还有149.87Hz~168.26Hz的低频带隙，但是带宽较窄，每条带隙的宽度只有20Hz~30Hz.当有一个方向压缩到第二稳态时，原来第三阶频率和第四阶频率之间的带隙消失，第五阶频率和第六阶频率之间出现另一条带隙，频率范围为57.88Hz~93.80Hz.当两个方向都压缩到第二稳态时，低频带隙发生了显著的变化，最低带隙的频率降低到22.16Hz~52.37Hz，第二条带隙变宽.通过上述研究，该多稳态超材料表现出良好的带隙调控性能.

  • 2.3 参数分析

  • 本节分析双稳态连杆厚度t及支撑结构宽度a2对带隙的影响.

  • 图4 双稳态连杆厚度t对结构带隙的影响

  • Fig.4 Effect of bistable connecting rod thickness t on structural band gap

  • 在双稳态连杆跨度和高度一定的情况下，厚度成为影响双稳态性能最主要的因素，因此本节分析其对带隙的影响.综合考虑几何尺寸的适配程度，以及产生大变形和多稳态的难易程度，选取t从1mm到2.5mm变化时对带隙的影响，间隔选为0.15mm.如图4所示为结构带隙数量、位置及宽度随着厚度t变化的分布情况.未变形结构[图4（a）]最低频率的带隙随t的增大而逐渐变窄且频率也逐渐增大，第二条带隙同样随t的增大逐渐上移，当t增大到1.75mm时该带隙消失.总体来说，随着t的增大，带隙向高频移动且逐渐变窄.单方向变形结构[图4（b）]的带隙随着t变化的趋势与未变形结构带隙相似，但会出现两条新的带隙.相比前两种情况而言，双方向变形结构[图4（c）]的两条带隙宽度更宽，频率范围更低，且随着t的增大而逐渐上移，并出现另一条窄带.

  • 支撑结构对结构刚度有很大的影响，因此分析其宽度对结构带隙的影响.综合考虑几何尺寸的适配程度及结构刚度，选取a2从2mm到14mm，间隔2mm.如图5所示为结构禁带数量、位置及宽度随着支撑结构宽度变化的分布情况.对于未变形结构[图5（a）]，随着a2的增大，0Hz~200Hz带隙从无到有且逐渐变宽.对于单方向变形结构[图5（b）]，当a2小于6mm时带隙主要在100Hz以上，当其逐渐增大时，出现了在100Hz以下的连续带隙.对于双方向变形结构[图5（c）]，当a2为2mm时，只有一条窄带.当a2逐渐增大时，带隙增多且变宽，且有逐渐合并的趋势.综上，当a2为6mm~10mm时，结构可以得到较宽范围的低频带隙.

  • 图5 支撑结构宽度a2对结构带隙的影响

  • Fig.5 Effect of width of support structure a2 on structural band gap

  • 3 传输率计算

  • 为了验证带隙的正确性，本节研究由有限个单胞组成的结构对振动的衰减能力.通过在一端施加位移激励，另一端对应位置测量其位移响应，可得到传输率.传输率的计算公式为：

  • T=20log⁡doutdin
    (5)

  • 其中din为激励位置的位移，dout为测量位置的位移.

  • 为了验证不同单胞数量对传输率的影响，分别计算了3×3、5×5、7×7和9×9四种不同结构在相同激励下的传输率，如图6所示.研究发现，不同单胞数量结构的传输率趋势一致，仅在数值上有所不同，单胞数量越多，传输率越小.综合考虑结构尺寸及计算时间成本等因素，后续选取5×5单胞组成的有限结构开展进一步研究.

  • 图6 具有不同数量单胞结构的传输率

  • Fig.6 The transmissibility of structures with different numbers of unit cells

  • 图7（a）、（c）和（e）分别为未变形结构、单方向变形结构和双方向变形结构，图7（b）、（d）和（f）分别为对应结构的传输率.可以看到未变形结构、单方向变形结构、双方向变形结构的带隙所对应的频率范围内振动皆有较大程度的衰减.总的来说，带隙和传输率可以较好地对应.

  • 为了进一步研究结构变形对振动衰减的影响，本节设计并计算了几种不同变形组合下结构沿变形方向的传输率，共有不变形、1列变形、3列变形、5列变形四种情况，计算结果如图8所示.分析发现，仅一列变形时，其低频范围内的传输特性即发生显著改变，显示出比未变形结构更强的衰减效果，且变形列数越多，衰减效果越强.通过上述分析说明，整体结构中局部一行或一列单胞变形即可在该方向上实现对应带隙频率范围内弹性波传播的衰减.另外，为了研究局部变形位置对振动衰减的影响，计算了结构中第1~5列分别变形时的传输率，其中靠近激励位置为第1列.结果如图9（b）所示，可以看出局部变形位置对振动衰减影响不大.

  • 图7 不同变形形式结构及对应传输率

  • Fig.7 The transmissibility of structures with different deformation

  • 图8 不同变形列数对传输率的影响

  • Fig.8 Effect of different number of deformed column on transmissibility

  • 图9 变形位置示意及其对传输率的影响

  • Fig.9 Schematic of deformation position and its effect on transmissibility

  • 4 结论

  • 本文设计了一种多稳态超材料，结合有限元和布洛赫定理，计算了结构的能带结构及有限尺寸结构的传输率，对其带隙特性及调控性能进行了研究.研究发现：

  • （1）双稳态连杆的变形行为可以显著改变结构低频范围内的带隙，表现出优异的带隙调控性能.

  • （2）通过参数分析得到了几何参数对结构带隙的影响规律，可根据需要设计具有低频宽带的几何结构.

  • （3）通过对有限尺寸结构传输率的计算验证了带隙的正确性，并分析验证了对输入波的衰减特性，发现整体结构中局部一行或一列单胞变形即可在该方向对应带隙频率范围内实现弹性波传播的衰减.

  • 参考文献

    • [1] ZHANG K,DENG Z C,MENG J M,et al.Wave propagation in hexagonal lattices with plateau borders [J].Composite Structures,2016,140:525-533.

    • [2] ZHANG K,HONG F,LUO J,et al.Topological insulator in a hexagonal plate with droplet holes [J].Journal of Physics D:Applied Physics,2020,54(10):105502.

    • [3] ZHANG K,LUO J,HONG F,et al.Seismic metamaterials with cross-like and square steel sections for low-frequency wide band gaps [J].Engineering Structures,2021,232:111870.

    • [4] ZHAO C,ZHANG K,ZHAO P C,et al.Elastic wave propagation in nonlinear two-dimensional acoustic metamaterials [J].Nonlinear Dynamics,2022,108(2):743-763.

    • [5] ZHAO P,ZHANG K,DENG Z C.Size effects on the band gap of flexural wave propagation in one-dimensional periodic micro-beams [J].Composite Structures,2021,271:114162.

    • [6] 李夏临,杨晓东,张伟.二维陀螺声子晶体结构力学研究 [J].动力学与控制学报,2020,18(4):79-85.LI X L,YANG X D,ZHANG W.Analysis of dynamics characteristics of 2-D gyroscope phononic crystal.Journal of Dynamics and Control [J].2020,18(4):79-85.(in Chinese)

    • [7] 王伟能,杨晓东,张伟.二维三角陀螺声子晶体的波调控研究 [J].动力学与控制学报,2021,19(1):75-79.WANG W N,YANG X D,ZHANG W.Study on wave manipulation in 2-D traingular gyroscope phononic crystals.Journal of Dynamics and Control [J].2021,19(1):75-79.(in Chinese)

    • [8] CELLI P,GONELLA S.Tunable directivity in metamaterials with reconfigurable cell symmetry [J].Applied Physics Letters,2015,106(9):091905.

    • [9] SCHAEFFER M,RUZZENE M.Wave propagation in multistable magneto-elastic lattices [J].International Journal of Solids and Structures,2015,56-57:78-95.

    • [10] BERTOLDI K,BOYCE M C.Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures [J].Physical Review B,2008,77(5):052105

    • [11] BERTOLDI K,BOYCE M C.Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations [J].Physical Review B,2008,78(18):184107.

    • [12] SHIM J,WANG P,BERTOLDI K.Harnessing instability-induced pattern transformation to design tunable phononic crystals [J].International Journal of Solids and Structures,2015,58:52-61.

    • [13] BOWEN A G,ZUCCO G,WEAVER P M.Morphing of symmetric cross-ply cylindrical shells by minimising the Brazier moment:Optimised hinge folding [J].Thin-Walled Structures,2021,158:107122.

    • [14] CHILLARA VSC,HEADINGS L M,TSURUTA R,et al.Shape memory alloy-actuated prestressed composites with application to morphing automotive fender skirts [J].Journal of Intelligent Material Systems and Structures,2018,30(3):479-494.

    • [15] LACHENAL X,DAYNES S,WEAVER P M.Review of morphing concepts and materials for wind turbine blade applications [J].Wind Energy,2013,16(2):283-307.

    • [16] QIU J,LANG J H,SLOCUM A H.A curved-beam bistable mechanism [J].Journal of Microelectromechanical Systems,2004,13(2):137-146.

    • [17] WALKER M G,SEFFEN K A.On the shape of bistable creased strips [J].Thin-Walled Structures,2018,124:538-545.

    • [18] ZHANG K,QI L Y,ZHAO P C,et al.Buckling induced negative stiffness mechanical metamaterial for bandgap tuning [J].Composite Structures,2023,304:116421.

    • [19] CHEN L,XU A,LOU J,et al.Transition waves in one-dimensional periodic bistable mass-spring chains [J].International Journal of Structural Stability and Dynamics,2022,22(9):2250100.

    • [20] LIBRANDI G,TUBALDI E,BERTOLDI K.Programming nonreciprocity and reversibility in multistable mechanical metamaterials [J].Nature Communications,2021,12(1):3454.

    • [21] NADKARNI N,ARRIETA A F,CHONG C,et al.Unidirectional transition waves in bistable lattices [J].Physical Review Letters,2016,116(24):244501.

    • [22] RAMAKRISHNAN V,FRAZIER M J.Transition waves in multi-stable metamaterials with space-time modulated potentials [J].Applied Physics Letters,2020,117(15):151901.

    • [23] MEAUD J,CHE K.Tuning elastic wave propagation in multistable architected materials [J].International Journal of Solids and Structures,2017,122-123:69-80.

    • [24] MEAUD J.Multistable two-dimensional spring-mass lattices with tunable band gaps and wave directionality [J].Journal of Sound and Vibration,2018,434:44-62.

    • [25] HU W,REN Z,WAN Z,et al.Deformation behavior and band gap switching function of 4D printed multi-stable metamaterials [J].Materials & Design,2021,200:109481.

    • [26] REN Z,JI L,TAO R,et al.SMP-based multi-stable mechanical metamaterials:From bandgap tuning to wave logic gates [J].Extreme Mechanics Letters,2021,42:101077.

    • [27] WANG J X,LIU X,YANG Q S,et al.A novel programmable composite metamaterial with tunable Poisson's ratio and bandgap based on multi-stable switching [J].Composites Science and Technology,2022,219:109245.

    • [28] ZHANG K,ZHAO P C,HONG F,et al.On the directional wave propagation in the tetrachiral and hexachiral lattices with local resonators [J].Smart Materials and Structures,2020,29(1):015017.

    • [29] TAN X,WANG B,WANG L,et al.Effect of beam configuration on its multistable and negative stiffness properties [J].Composite Structures,2022,286:115308.

  • 基本信息

    中图分类号: O327
    文献标识码: A
    DOI: 10.6052/1672-6553-2023-030
    文章编号: 1672-6553-2023-21(7)-005-007

    基金信息

    国家自然科学基金资助项目(12172297)；
    大连理工大学工业装备结构分析国家重点实验室开放基金资助项目(GZ22106)；
    National Natural Science Foundation of China (12172297)；
    the Open Foundation of State Key Laboratory of Structural Analysis for Industrial Equipment (GZ22106)；

    引用信息

    稿件历史

    收稿日期: 2023-02-15

  • 参考文献

    • [1] ZHANG K,DENG Z C,MENG J M,et al.Wave propagation in hexagonal lattices with plateau borders [J].Composite Structures,2016,140:525-533.

    • [2] ZHANG K,HONG F,LUO J,et al.Topological insulator in a hexagonal plate with droplet holes [J].Journal of Physics D:Applied Physics,2020,54(10):105502.

    • [3] ZHANG K,LUO J,HONG F,et al.Seismic metamaterials with cross-like and square steel sections for low-frequency wide band gaps [J].Engineering Structures,2021,232:111870.

    • [4] ZHAO C,ZHANG K,ZHAO P C,et al.Elastic wave propagation in nonlinear two-dimensional acoustic metamaterials [J].Nonlinear Dynamics,2022,108(2):743-763.

    • [5] ZHAO P,ZHANG K,DENG Z C.Size effects on the band gap of flexural wave propagation in one-dimensional periodic micro-beams [J].Composite Structures,2021,271:114162.

    • [6] 李夏临,杨晓东,张伟.二维陀螺声子晶体结构力学研究 [J].动力学与控制学报,2020,18(4):79-85.LI X L,YANG X D,ZHANG W.Analysis of dynamics characteristics of 2-D gyroscope phononic crystal.Journal of Dynamics and Control [J].2020,18(4):79-85.(in Chinese)

    • [7] 王伟能,杨晓东,张伟.二维三角陀螺声子晶体的波调控研究 [J].动力学与控制学报,2021,19(1):75-79.WANG W N,YANG X D,ZHANG W.Study on wave manipulation in 2-D traingular gyroscope phononic crystals.Journal of Dynamics and Control [J].2021,19(1):75-79.(in Chinese)

    • [8] CELLI P,GONELLA S.Tunable directivity in metamaterials with reconfigurable cell symmetry [J].Applied Physics Letters,2015,106(9):091905.

    • [9] SCHAEFFER M,RUZZENE M.Wave propagation in multistable magneto-elastic lattices [J].International Journal of Solids and Structures,2015,56-57:78-95.

    • [10] BERTOLDI K,BOYCE M C.Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures [J].Physical Review B,2008,77(5):052105

    • [11] BERTOLDI K,BOYCE M C.Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations [J].Physical Review B,2008,78(18):184107.

    • [12] SHIM J,WANG P,BERTOLDI K.Harnessing instability-induced pattern transformation to design tunable phononic crystals [J].International Journal of Solids and Structures,2015,58:52-61.

    • [13] BOWEN A G,ZUCCO G,WEAVER P M.Morphing of symmetric cross-ply cylindrical shells by minimising the Brazier moment:Optimised hinge folding [J].Thin-Walled Structures,2021,158:107122.

    • [14] CHILLARA VSC,HEADINGS L M,TSURUTA R,et al.Shape memory alloy-actuated prestressed composites with application to morphing automotive fender skirts [J].Journal of Intelligent Material Systems and Structures,2018,30(3):479-494.

    • [15] LACHENAL X,DAYNES S,WEAVER P M.Review of morphing concepts and materials for wind turbine blade applications [J].Wind Energy,2013,16(2):283-307.

    • [16] QIU J,LANG J H,SLOCUM A H.A curved-beam bistable mechanism [J].Journal of Microelectromechanical Systems,2004,13(2):137-146.

    • [17] WALKER M G,SEFFEN K A.On the shape of bistable creased strips [J].Thin-Walled Structures,2018,124:538-545.

    • [18] ZHANG K,QI L Y,ZHAO P C,et al.Buckling induced negative stiffness mechanical metamaterial for bandgap tuning [J].Composite Structures,2023,304:116421.

    • [19] CHEN L,XU A,LOU J,et al.Transition waves in one-dimensional periodic bistable mass-spring chains [J].International Journal of Structural Stability and Dynamics,2022,22(9):2250100.

    • [20] LIBRANDI G,TUBALDI E,BERTOLDI K.Programming nonreciprocity and reversibility in multistable mechanical metamaterials [J].Nature Communications,2021,12(1):3454.

    • [21] NADKARNI N,ARRIETA A F,CHONG C,et al.Unidirectional transition waves in bistable lattices [J].Physical Review Letters,2016,116(24):244501.

    • [22] RAMAKRISHNAN V,FRAZIER M J.Transition waves in multi-stable metamaterials with space-time modulated potentials [J].Applied Physics Letters,2020,117(15):151901.

    • [23] MEAUD J,CHE K.Tuning elastic wave propagation in multistable architected materials [J].International Journal of Solids and Structures,2017,122-123:69-80.

    • [24] MEAUD J.Multistable two-dimensional spring-mass lattices with tunable band gaps and wave directionality [J].Journal of Sound and Vibration,2018,434:44-62.

    • [25] HU W,REN Z,WAN Z,et al.Deformation behavior and band gap switching function of 4D printed multi-stable metamaterials [J].Materials & Design,2021,200:109481.

    • [26] REN Z,JI L,TAO R,et al.SMP-based multi-stable mechanical metamaterials:From bandgap tuning to wave logic gates [J].Extreme Mechanics Letters,2021,42:101077.

    • [27] WANG J X,LIU X,YANG Q S,et al.A novel programmable composite metamaterial with tunable Poisson's ratio and bandgap based on multi-stable switching [J].Composites Science and Technology,2022,219:109245.

    • [28] ZHANG K,ZHAO P C,HONG F,et al.On the directional wave propagation in the tetrachiral and hexachiral lattices with local resonators [J].Smart Materials and Structures,2020,29(1):015017.

    • [29] TAN X,WANG B,WANG L,et al.Effect of beam configuration on its multistable and negative stiffness properties [J].Composite Structures,2022,286:115308.

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