REVIEW OF LOW-FREQUENCY ELASTIC WAVE METAMATERIALS
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摘要: 超材料是一类新兴的具有超常物理性质的人造周期/拟周期材料, 能够改变电磁波、声波以及弹性波等在介质中的传播特性. 因在航天、国防以及民用科学等方面的巨大应用潜力, 超材料自被提出后便受到极大的关注并引发研究热潮. 弹性波超材料是超材料的一种, 能够基于弹性波与超材料结构的相互耦合作用实现对弹性波的操控. 带隙是评估弹性波超材料实现弹性波操控的重要工具, 其性质与超材料的材料参数、晶格常数以及局域振子的固有频率相关. 受制于超材料的承载能力、外观尺寸以及局域振子结构等因素, 利用传统超材料开启低频(约100 Hz)弹性波带隙依然存在较大困难. 文章首先简要介绍超材料开启弹性波带隙的基本原理, 然后从低频弹性波超材料基本结构与低频带隙实现方法、低频带隙优化与调控策略、低频带隙潜在应用等三个方面详细总结低频弹性波超材料的研究工作. 其中, 低频带隙超材料的基本结构主要包括布拉格散射型超材料、传统局域共振型超材料以及准零刚度局域共振超材料. 文章通过总结低频弹性波超材料的研究进展, 分析了目前研究中的不足并对未来低频弹性波的研究方向进行了展望.Abstract: Metamaterial, a type of burgeoning man-made material/structure, possesses a periodic/quasi-periodic structure and is able to change the transmission properties of the electromagnetic wave, the acoustic wave and the elastic wave. Due to its enormous potentiality in the field of the spaceflight, national defence and civilian, the metamaterial attracted great interest, inspired a new wave of research and obtained consecutive important achievement since it was proposed. Elastic wave metamaterial is a kind of metamaterial which is capable of realizing the attenuation and manipulation of the elastic wave on the basis of the interaction of the elastic wave and the periodic/ quasi-periodic structure. Band structure design is an important tool for the elastic wave metamaterial to execute the wave manipulation and attenuation. The location, width and wave suppression performance of the frequency band are related to the nature of materials, the lattice constant of the metamaterial, and the resonant frequency of the local resonator. Because of the limitations such as the carrying capacity, the overall size, and the structure of the local resonator, it is still difficult to obtain an elastic wave band gap in the frequency range around 100 Hz through the conventional metamaterials. This review introduces the fundamental principle of the metamaterial for opening elastic wave band gaps firstly, and then elaborates the low-frequency elastic wave metamaterial from three aspects: the fundamental configuration of the metamaterial, the low-frequency band gap optimization and tuning, and some potential applications. The fundamental configurations of low-frequency elastic wave metamaterials mainly include three aspects: Bragg scattering metamaterials, conventional local resonant metamaterials and quasi-zero-stiffness local resonant metamaterials. The low-frequency band tunability achieved by both the passive and active approaches detailed as well. This review summarizes the current knowledge of the low-frequency elastic wave metamaterial, analyzes the inadequacies and the advantages in current research, and outlines future research prospects.
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引 言
超材料(metamaterial)是指人为构造的一类材料或结构, 能够实现传统自然界材料所不能实现的一些性质或功能, 比如带隙结构[1]、波导[2]、负泊松比[3]、负折射率[4]、负膨胀系数[5]、负质量[6]、超透镜[7]、声学隐身[8]以及拓扑态[9]等. 显然, 超材料的奇异性质超越了自然规律的限制, 使其获得了与自然界中材料迥然不同的超常物理性质, 极大地拓展了材料的应用范围, 并引发了诸如新一代信息技术、国防工业、新能源技术、细微加工技术等领域的重大变革, 获得了包括政府、学术界以及产业界的极大关注[10-12].
美国国防部将超材料列为“六大颠覆性基础研究领域”之一, 并开发出能够适用于航天飞机隔热设备的具有复原功能的反弹陶瓷管[13]. 德国的研究人员开发出能够让飞机在雷达探测范围内隐身的隐身超材料. 而日本和俄罗斯也将超材料作为战斗机隐身的核心技术, 并投入大量科研经费进行持续攻关. 法国科学家利用超材料的基本思想, 提出通过在地面上打孔的方式以削弱地震波对建筑物的影响. 荷兰科学家则发明了可编程智能橡胶, 其材料性质根据挤压程度而发生相应变化[14]. 在国家的大力支持以及科研、产业人员的共同努力下, 我国在局域共振超材料理论[15]、薄膜超材料[16]、电磁超材料[17]、负属性超材料[18-19]以及非线性超材料[20-21]等领域取得了若干原创性研究成果, 并始终处在领先位置. 此外, 在电磁黑洞、声学黑洞和声学隐身等功能性超材料研制方面, 我国科研工作者也颇有建树[22-23]. 我国以光启科学为首的超材料研发制造企业的迅猛发展标志着超材料在我国正加速从理论研究转向工程应用.
一般而言, 超材料包括基体材料和散射体材料等两类材料. 散射体材料通过一定规律嵌入/放置在基体材料中. 散射体材料的形式多种多样, 包括金属、液体、塑料甚至是空气[24]. 需要指出的是, 超材料所呈现的奇异物理性质既不取决于基体材料, 也不取决于散射体材料, 而是由二者组合而成的新结构所决定[25]. 超材料大致可以分为电磁波超材料、声学超材料、弹性超材料、结构超材料等[26-27]. 当波通过超材料时, 超材料通过阻碍[28]、吸收[29]以及集聚[30]等方式, 实现了对不同类型的波的操控.
当波在超材料中传播时, 波与超材料结构相互作用(散射、反共振等), 从而产生能够抑制波传播的带隙结构[25]. 对于能够抑制弹性波传播的超材料带隙而言, 其形成机理主要分为固体中横波和纵波之间的波形转化与相互作用(布拉格散射带隙)以及局域振子的反共振(局域共振带隙)等两种[24]. 其中, 布拉格散射带隙的带隙频率与超材料的晶格常数以及超材料构成组分的材料参数相关, 而局域共振带隙的带隙频率依赖于局域振子的固有频率[31]. 详细来讲, 随着超材料晶格常数减小或基体材料弹性模量变大, 布拉格散射型带隙频率明显向高频区域移动[32], 换句话说, 难以使用具有支撑能力的小尺度超材料在低频范围内开启带隙结构. 幸运的是, 局域共振带隙的中心频率仅与局域振子的固有频率相关, 而机械系统的固有频率属于机械系统的基本属性, 依赖于系统的惯性与刚度特性[33]. 因此, 局域共振带隙机制的提出, 打破了超材料晶格常数与基体材料参数对带隙频率的限制, 为使用具有支撑能力的小尺度超材料开启低频带隙提供了契机[15].
然而, 传统局域振子存在大承载能力与小刚度之间的矛盾, 局域振子固有频率依然较高, 超材料难以在更低频率范围( < 100 Hz)内开启弹性波带隙结构[34], 阻碍了超材料在低频振动控制、低频振动能量俘获、低频波导以及低频逻辑结构设计等方面的应用. 尽管获得低频弹性波带隙的理论基础明晰, 但如何创新设计超材料结构, 以实现超材料的功能性与实用性, 依然制约着低频超材料的发展. 本文拟从超材料带隙结构开启机理出发, 从超材料结构设计、带隙频率降低方法、低频带隙优化与调控策略等方面入手, 系统阐述在保证超材料承载能力以及外观尺寸等实用性的前提下开启低频弹性波带隙的基本方法, 总结低频弹性波超材料的潜在应用.
1. 基本方法与结构
布拉格散射和局域共振是超材料获得弹性波带隙的两种基本机制. 然而, 无论是布拉格散射型带隙, 还是局域共振型带隙, 其带隙频率都和超材料的基本结构严格相关. 换句话说, 设计合理的超材料结构, 是获得低频弹性波带隙的关键. 本节旨在总结低频弹性波( < 100 Hz)超材料的基本结构并对其进行分类, 以厘清低频弹性波带隙与超材料构型之间的关系.
1.1 布拉格散射型超材料
当超材料的基体为流体或者气体时, 基体仅能传播纵波. 因为超材料相邻元胞间反射波的同相叠加, 部分频率的弹性波在超材料结构中传播时将被削弱, 形成带隙结构, 其中心频率为[25]
$$ {\omega _{\text{c}}} = \frac{c}{{2a}} $$ (1) 式中, c与a分别表示弹性波的波速以及超材料结构的晶格常数. 因基体中弹性波的波速与基体的弹性模量呈正相关, 所以减小基体弹性模量可以有效降低布拉格散射型带隙的中心频率, 在低频区域形成带隙结构[35]. 此外, 增大超材料结构的晶格常数, 也能够降低带隙的中心频率.
当超材料的基体材料为固体时, 基体中存在横波和纵波之间的波形转化. 单个散射体在米氏散射(Mie scattering)过程中低频纵波向横波的转化是布拉格散射型带隙形成的基本原因. 需要注意的是, 该波形转化主要取决于一阶谐波的米氏散射, 且在散射峰值附近, 散射体呈刚体共振模式[36]. 若将周期性超材料简化为弹簧−质量原子链(其中基体简化为弹簧和质量块, 散射体为质量块), 其起始频率与截止频率分别表示为[37]
$$ {\omega _{\text{B}}} \propto \sqrt {\dfrac{{2k}}{M}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\omega _{\text{E}}} \propto \sqrt {\dfrac{{2k}}{m}} $$ (2) 式中, m和M分别表示两种简化质量块的质量; k表示连接两个质量块的弹簧的刚度. 显然, 布拉格散射型带隙的带隙位置与散射体材料和基体材料的材料参数相关.
对于布拉格散射超材料而言, 增大散射体材料密度、降低基体材料弹性模量以及增大晶格常数均可降低带隙起始频率, 在低频范围内获得弹性波带隙结构[38]. 然而, 增大晶格常数会导致超材料整体尺寸明显增大, 严重制约超材料在工程实际中的应用. 而降低基体弹性模量或者增大散射体密度, 即减小式(2)中的k值或者增大M值, 散射体的刚体共振频率减小, 布拉格散射型带隙起始频率向低频范围移动, 能够在低频范围内获得弹性波带隙.
通过类比DNA结构, 文献[39]提出了一种一维超材料结构. 该超材料由圆形板和螺旋结构组成, 其中螺旋结构包括两侧的螺旋线和连接螺旋线的中间连接杆. 将圆形板简化为质量块, 螺旋结构简化为弹簧, 则该DNA超材料可以等效为传统弹簧−质量原子链结构. 改变螺旋线之间的连接杆数量, 可以在较大范围内对螺旋结构的刚度进行调节, 这为在低频范围内开启带隙结构提供了潜在途径. 但是, 当超材料的刚度过低时, 超材料承受静载的能力下降, 其工程应用潜力降低.
如果不考虑超材料的承载能力, 即超材料不承受静载时, 设计无支撑能力的超材料能够在低频范围内获得弹性波带隙. 如图1(a)所示, 通过在两个超材料元胞的连接弹簧中间安装旋转结构, 可以消除连接弹簧的变形, 使纵向刚度等于0. 该纵向刚度等于0的超材料的色散关系可以表示为[40]
$$ {M_{{\text{eff}}}}{\omega ^2} = 4{K_{{\text{eff}}}}{\sin ^2}\left( {qD/2} \right) $$ (3) 式中,
$ {M_{{\text{eff}}}} $ 和$ {K_{{\text{eff}}}} $ 分别表示该超材料的等效质量和等效刚度; q和D分别表示波数和超材料的晶格常数. 从超材料的色散曲线得出: 该纵向刚度等于0的超材料在频率区间$\left[ {0,{\kern 1 pt} {\kern 1 pt} \sqrt {2 k/m} /(2{\text{π}})} \right]\;{\text{Hz}}$ 内形成了布拉格散射带隙.构造无支持能力超材料并实现低频带隙的第二种方法为通过铰链连接质量块. 由于铰链的引入, 超材料的转动刚度等于0. 由铁木辛科梁理论可知, 弯曲波由材料的垂向运动与旋转运动等两种运动共同控制[41]. 对于传统的梁结构而言, 能够在低频范围内产生两支色散曲线, 其中一支从0开始, 另外一支从特定的起始频率开始. 因两支色散曲线分布于整个低频区域, 所以传统梁结构不能在低频范围内开启带隙结构[42].
将传统梁结构色散方程中的转动刚度置为0并求解色散关系, 可以获得转动刚度等于0的超材料的色散曲线表达式为[41]
$$ \left. \begin{array}{l}\omega \text{ = 0,}\;\;{\rm{1st}}\;\;{\rm{branch}}\\ \omega = \sqrt {\dfrac{{\alpha a{k^2}}}{{\rho {A_{\rm b}}}} + \dfrac{{\alpha a}}{{\rho {I_{\rm b}}}}} ,\;\;\rm 2nd\;\;\rm branch \end{array}\right\}$$ (4) 由式(4)可知, 当转动刚度等于0时, 超材料的其中一支色散曲线与波数轴重合, 即对于任意波数k, 其对应的频率均等于0. 因此, 该超材料能够开启起始频率为0的低频弹性波带隙结构.
无支撑能力超材料的第三种实现方式如图1(c)所示. 该类超材料主要通过切削连续均质等截面梁形成. 切削处梁截面逐渐减小, 形成锥型结构. 因锥型体最小截面处截面面积远远小于未切削均质连续体的截面面积, 所以最小截面处的弯曲刚度与扭转刚度均非常小, 这为实现低频布拉格散射带隙提供了可能. 此外, 通过超材料中局部机构位移与整体超低频波动强耦合以及在超材料中引入几何非线性等方式, 也能够开启起始频率为0的超低频弹性波带隙[44-46].
五模超材料也称为“金属流体”, 能够对纵波和横波进行解耦, 其体积模量B与剪切模量G的比值远远大于自然界固体材料的比值[47-49]. 基于五模超材料的特殊物理性质, 北京理工大学胡更开教授团队首次实现了水声完美隐身[50]. 此外, 通过在五模超材料中引入负泊松比微结构设计, 该团队还发现了一种“反流体”的波动特性, 并实现了弹性波的宽频极化控制[51-52]. 由五模超材料所开启带隙的中心频率与材料参数的关系可以表示为[53]
$$ {\omega _{\text{c}}} = A - B \ln \left( {E/\rho + C} \right) $$ (5) 式中, A和B表示两个负实数, C表示正实数. E与
$\rho $ 分别表示材料的弹性模量和密度. 显然, 减小弹性模量或增大材料质量密度均可以降低五模超材料的带隙中心频率, 在低频范围内开启弹性波带隙结构[54]. 例如, 将五模超材料的构成材料由铝置换成橡胶, 带隙范围由812~6430 Hz降低至1.48~11.5 Hz[53]. 此外, 使用密度较大的材料与橡胶材料形成复合五模超材料结构, 能够增大五模超材料的等效质量密度, 进一步降低带隙所处的频率范围[55-56].值得注意的是, 不管是由基体材料与散射体构成的传统超材料, 还是五模超材料等新型超材料, 较大晶格常数以及较低材料刚度属性是获得低频带隙的两个关键条件[25]. 然而, 晶格常数增加会明显扩大超材料的外观尺寸, 使其应用潜力下降; 降低材料刚度会影响超材料的静载承载能力, 使其成为无支撑能力结构[24]. 因此, 创新设计具有承载能力但刚度较小的布拉格散射型超材料, 是实现利用超材料进行低频振动控制的理想途径.
Zhou等[57]结合打靶法, 率先设计了一种可增材制造的一体化超材料结构, 其示意图如图2(a)所示. 由多段梁组成的弹性连接部分的刚度与施加在该超材料两端的静载相关, 即超材料刚度随屈曲梁压缩量的变化而变化[58]. 当施加静载等于额定静载时, 超材料刚度趋近于0. 与图1所示的无支撑超材料不同的是, 准零刚度链式超材料在静载条件下出现零刚度特性, 即超材料具有良好的承受静载的能力[59]. 图2(b)表示零刚度链式超材料的带隙结构. 显然, 在一定压缩量范围内, 超材料带隙频率随超材料压缩量的增大而减小, 换句话说, 零刚度链式超材料能够在低频(约10 Hz)范围开启带隙结构.
1.2 传统弹性元件局域共振超材料
2000年, Liu等[15]在Science上发文, 首次提出了局域共振超材料的概念. 局域共振超材料的带隙形成机理与布拉格散射型超材料的带隙形成机理具有本质区别, 主要基于安装在基体结构上的局域共振单元的强共振特性开启带隙机构[14, 60-61]. 局域共振带隙的中心频率近似等于局域振子的共振频率. 因此, 创新设计局域振子结构, 使其具有低固有频率特性, 是利用小尺度超材料开启低频弹性波带隙的核心.
一般而言, 由线性弹簧和质量块组成的如图3(a)所示的弹簧−质量局域共振子, 是构成局域共振超材料最简单的局域共振子构型. 利用该局域振子, 能够分析局域共振带隙的形成机理. 将弹簧−质量局域共振子周期性地安装在弹簧−质量原子链[62]、梁[63]以及板[64]上, 可形成一维和二维局域共振超材料的原理模型, 并分析纵波和弯曲波带隙的形成机理. 研究发现, 当弹性波频率靠近局域振子的固有频率时, 局域振子中质量块的动力学响应幅值增大, 弹性波在主体结构中的传播幅值减小, 超材料形成弹性波带隙结构[65]. 局域共振带隙的起始与终止频率可以由下式给出, 即[34]
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$$ {\omega _{\text{B}}} \propto \sqrt {\frac{{{k_{\text{R}}}}}{{{m_{\text{R}}}}}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\omega _{\text{E}}} \propto {\omega _{\text{B}}}\sqrt {1 + \frac{{{m_{\text{R}}}}}{M}} $$ (6) 式中, kR和mR分别表示局域振子的刚度和质量; M表示超材料元胞中基体的质量. 显然, 局域共振带隙的频率与宽度均与局域振子的属性相关, 打破了布拉格散射带隙对超材料晶格常数以及基体材料属性的依赖, 为利用小尺度且具有支撑能力的基体结构在低频区域内开启带隙结构提供了潜在途径. 然而, 就线性弹簧−质量块局域振子而言, 其外观尺寸较大, 且线性弹簧存在承载和低刚度之间的矛盾, 导致该局域振子很难应用于工程实际. 因此, 开发新型局域振子结构, 是推动超材料由理论研究转向工程应用的必然趋势.
通过弹性材料或者弹性结构替代线性弹簧, 可以构造紧凑型局域振子. 如图3所示, 用于设计局域振子的弹性材料包括橡胶[66-72]、薄膜[73-74]、悬臂梁[75]以及聚合物混凝土[76]等. 使用弹性结构替代线性弹簧的一个优势在于弹性材料可以同时在多个方向上提供回复力. 因此, 由弹性结构构成的局域振子能够开启不同类型的弹性波带隙, 例如纵波带隙、弯曲波带隙以及扭转波带隙等[77]. 需要指出的是, 若使用球摆类结构构造超材料元胞, 也可以在同一超材料中获得多类型低频弹性波带隙, 从而实现对不同种类的低频弹性波进行控制[78].
除开弹性结构外, 将弹性基体进行切削镂空以形成特定的几何结构, 也是构造紧凑型局域振子的一种方法[79-84]. 因连接局域振子质量的基体被镂空, 如图3(c)所示, 其刚度远远小于完整连续体的刚度, 局域振子的固有频率较低, 超材料能够在低频范围内开启弹性波带隙结构. 然而, 无论是由线性弹簧和质量块组成的传统型局域振子, 亦或是由弹性材料或弹性结构以及质量块组成的紧凑型局域振子, 都存在支撑能力与低刚度之间的矛盾. 换句话说, 当局域振子质量块质量较大时, 连接质量块的弹性元件需要较大的刚度以支撑质量块, 局域振子固有频率难以降低. 由于局域共振型超材料的带隙频率与局域振子的属性相关, 所以由普通弹性元件和质量块组成的局域共振超材料难以在低频尤其是超低频范围内开启弹性波带隙.
1.3 准零刚度局域共振超材料
固有频率是机械系统的固有属性, 与系统刚度和惯性相关[88]. 系统刚度一般用以评估系统抵抗变形的能力. 当系统变形增大, 所施加的力也增加时, 此时称系统具有正刚度特性. 相反, 当系统变形增大, 所施加的力减小时, 称系统具有负刚度特性. 将正负刚度并联组合, 利用负刚度机构的负刚度特性抵消正刚度元件的刚度值, 则组合系统的刚度值较正刚度系统刚度值明显减小[89]. 通过参数设计, 正负刚度并联组合系统的刚度值在静平衡位置处等于0, 在静平衡位置周围趋近于0. 此时, 该组合系统称为准零刚度系统, 其静力学特性如图4所示[90-91].
设计具有低固有频率的局域振子是利用超材料开启低频弹性波带隙的核心. 但是, 单一弹性元件难以达到承受静载与降低动刚度之间的统一, 导致所构成的局域振子固有频率较高, 难以开启低频弹性波带隙[89-93]. 准零刚度系统的提出, 为创新设计低频局域振子提供了契机. 2017年, Zhou等[34]率先提出准零刚度局域振子的概念, 并基于准零刚度局域振子设计了准零刚度局域共振超材料梁. 准零刚度局域振子的刚度−位移表达式可以表示为
$$ {K_{{\text{QZS}}}}\left( x \right) = {k_{\text{p}}} - \left( {1 - \gamma } \right)P\left( x \right) $$ (7) 式中, kp表示局域振子中正刚度元件的刚度值;
$\gamma $ 表示局域振子在静平衡位置处的刚度值与正刚度元件刚度值kp的比值, 其大小由负刚度机构的参与程度决定; P(x)表示与运动位移相关的变量. 显然, 准零刚度局域振子的刚度值可以通过改变$\gamma $ 的大小, 亦即负刚度机构的参与程度来进行调节.对于准零刚度局域振子而言, 负刚度机构是决定其外观尺寸和力学特性最核心的部分. 目前, 包括斜弹簧机构[93-97]、X型机构[98]、半球−半球机构[99]、凸轮−滚子机构[100]以及永磁铁环[101]等机械结构均可以实现负刚度特性并抵消正刚度元件(主要由线性螺旋弹簧和橡胶块组成)的刚度值以实现准零刚度特性, 如图5(a)所示. 但是, 由于正刚度机构与负刚度机构均由机械结构组成, 准零刚度局域振子外观尺寸较大, 自重较重, 导致所构成的准零刚度局域共振超材料尺寸较大. 此外, 正负刚度机构并联组合会引入难以避免的机械接触, 引起机械摩擦, 影响超材料带隙结构的振动抑制性能.
为了突破机械准零刚度局域振子的缺陷, Cai等[92]利用两对折叠梁和两对屈曲梁, 设计了如图5(b)-I所示的可快速增材制造的结构化准零刚度局域振子. 在该准零刚度局域振子中, 折叠梁为局域振子提供正刚度, 屈曲梁提供负刚度, 折叠梁与屈曲梁直接相连, 避免了冗余的机械接触, 减小了机械摩擦对超材料带隙结构的影响. 另外, 相较于机械准零刚度局域振子, 结构化准零刚度局域振子质量更轻, 整体尺寸更小, 更有利于设计紧凑型准零刚度局域共振超材料.
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尽管由折叠梁和屈曲梁构成的准零刚度局域振子已经解决了机械式准零刚度局域振子的诸多缺陷, 但是该局域振子依然由正负刚度元件组成, 结构依然较为复杂. 随着对结构化准零刚度局域共振子结构的持续性改进, 正刚度元件和负刚度元件融为一体, 改由多段柔性梁实现准零刚度结构[57], 如图5(b)-II所示. 更为重要的是, 该型准零刚度局域振子的静力学特性与多段梁的压缩量相关, 这为调控局域共振超材料的带隙结构提供了理论支撑[103].
由式(7)可知, 机械准零刚度局域振子的刚度值与负刚度机构的参与程度, 亦即刚度系数
$\gamma $ 相关[34]. 而结构化准零刚度局域振子的静力学特性则与多段梁的压缩量相关[103-105]. 因局域共振超材料的带隙频率与局域振子的刚度相关, 所以局域振子的参数对准零刚度局域共振超材料的带隙具有明显影响. 图6表示刚度系数与多段梁压缩量对准零刚度超材料带隙结构的影响. 显然, 随着刚度系数的减小, 即负刚度机构参与程度增加, 准零刚度超材料的带隙频率向低频区域移动, 这与多段梁压缩量对准零刚度超材料带隙结构的影响完全一致, 即随着多段梁压缩量的增加, 超材料带隙频率向低频区域移动. 综上所述, 准零刚度超材料能够在保证承载能力的前提下, 在低频区域内开启弹性波带隙.2. 低频带隙优化与调控方法
增大布拉格散射型超材料的晶格常数或改变构成超材料的基体与散射体的材料参数, 能够在低频范围内开启低频弹性波带隙结构[106]. 但是, 无论是增大超材料的晶格常数, 还是降低超材料基体材料的刚度, 均会制约超材料的工程应用. 局域共振型超材料的提出, 打破了布拉格散射带隙的带隙频率与晶格常数、基体材料参数之间的依赖关系, 成功利用具有承载能力的小尺度超材料结构开启了低频弹性波带隙[107]. 然而, 随着局域共振带隙频率向低频区域移动, 带隙宽度变窄, 带隙内弹性波衰减效果减弱[108]. 此外, 由于大部分超材料都是由不具备调控功能的机械结构组成, 低频带隙不易调控, 导致超材料的应用潜力下降. 因此, 发展低频带隙优化与调控方法, 对推进超材料的研究具有重要意义.
2.1 低频弹性波带隙拓宽与弹性波衰减性能优化
将低频范围内的单个带隙拓宽以及在低频范围内开启多个带隙是在低频范围内获得超宽弹性波带隙的两种主要方法. 就局域共振型超材料而言, 其带隙结构与超材料元胞内局域振子的固有频率相关. 因此, 创新设计局域振子结构或增加局域振子的自由度, 均可以拓宽低频带隙宽度. 目前, 设计多自由度局域振子[109-113]、在局域振子中引入惯容器[114-118]、在不同元胞的局域振子间建立耦合[119-123]以及通过依次改变质量或刚度的方式建立梯度局域振子[124-126]是拓宽低频弹性波带隙的主要方式. 当在局域振子中加入惯容器时, 局域振子的等效质量可以表示为[115]
$$ m_{{\text{eff}}}^{{\text{PI}}} = {m_{{\text{st}}}} - \frac{{m_2^2}}{{{m_2} + \left( {J - \dfrac{{{m_2}}}{{{\varOmega ^2}}}} \right)}} $$ (8) 式中,
$ {m_{{\text{st}}}} $ 与m2分别表示元胞的质量与局域振子的质量; J表示惯容系数. 在局域振子中加入惯容器可以明显扩大等效质量小于0的频率区间, 即低频带隙宽度增加.当增加局域振子的自由度时, 超材料的传递率曲线在低频区域内出现了多个衰减峰值, 即超材料在多个频率区域内开启了带隙结构, 在更大的频率范围内对作用于超材料上的弹性波起到了衰减作用[127]. 此外, 增加局域共振超材料中局域共振子的个数[128]以及在超材料中引入阻尼[129]也可以拓宽低频范围内带隙的宽度.
对超材料低频带隙进行优化的第二个方面为增强超材料在低频范围内对弹性波的衰减效果. 目前, 增强超材料带隙内弹性波的衰减性能以及削弱超材料通带频率区域的弹性波共振峰值是两种优化超材料弹性波衰减性能的主要方法. 就增强超材料带隙范围内弹性波衰减效果而言, 增大局域振子质量块的质量[92]和局域振子数量[100]是两种潜在的途径. 然而, 超材料在低频区域内所形成的带隙频率范围较低频区域内的通带频率范围较小, 也就是说, 对一般超材料而言, 通带频率区域在低频区域的占比较大[130]. 此外, 当弹性波频率处于通带内时, 弹性波响应幅值被放大, 不利于弹性波控制. 因此, 对通带内的弹性波传递性能进行优化(削减峰值), 能够有效提升超材料在低频范围内整体的弹性波衰减性能.
研究发现, 将传统局域振子中的线性弹簧更换为非线性弹簧所构成的非线性超材料, 能够产生桥接耦合现象[131]. 具体来说, 该类非线性超材料能够衰减带隙频率范围外通带区域内的弹性波传递幅值, 这样的通带区域被称为混沌带隙[21, 132-135]. 显然, 混沌带隙的提出, 为利用超材料在超宽频率范围操控弹性波提供了潜在途径, 具有较为重要的工程应用价值. 然而, 截止到目前, 由非线性超材料桥接耦合效应诱发的混沌带隙与外激励幅值相关, 限制了该类超材料的工程应用. 为打破桥接耦合效应受外激励幅值的限制, 研究学者通过双级谐振线性超材料也成功实现了桥接耦合效应, 在宽频带内实现了对弹性波的控制[136].
2.2 低频带隙调控方法
局域共振型超材料的带隙频率与局域振子的固有频率相关, 而固有频率属于局域振子的固有属性, 取决于惯性属性(质量、转动惯量)与刚度属性(垂向刚度、扭转刚度)[88]. 因此, 局域共振型低频带隙的调控方法主要包括基于局域共振子惯性的调控方法与基于局域共振子刚度的调控方法等两种.
目前, 在局域振子中引入惯性放大机构以及通过主动调控局域振子的质量是改变局域振子惯性属性来调控局域共振低频带隙的两种主要方法. 惯性放大机构能够产生惯性放大效应, 能够通过小质量产生较大惯性力[137]. 以弹簧−质量系统分析惯性放大机构, 获得惯性放大机构的等效质量为[138]
$$ {m_{{\text{eff}}}} = 2m + \frac{{{m_{\text{a}}}}}{{{{\tan }^2}\theta }} $$ (9) 式中, m和ma分别表示主结构质量和惯性放大机构的质量;
$\theta $ 表示惯性放大机构的刚性连接杆与主结构之间的夹角. 显然, 当夹角$\theta \in \left( {0,\text{π} /4} \right)$ 时, 引进惯性放大机构的机械系统的等效质量大于系统中质量块的质量之和, 换句话说, 该机械系统的惯性力通过机械结构被放大了.将如图7所示的由惯性放大机构组成的元胞进行周期性排布, 获得了惯性放大超材料[139-141]. 在系统反共振频率周围, 惯性放大机构产生的惯性力能够抵消主系统的弹性力, 所以惯性放大机构能够在低频范围内开启弹性波带隙结构[142]. 更为重要的是, 通过改变惯性放大机构的几何参数, 能够在较大范围内对系统的惯性力进行调节, 这为调控超材料的带隙位置提供了潜在途径.
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惯性放大机构除单独构成元胞以形成超材料外, 还能够同准零刚度局域振子进行结合以形成具有惯性放大效应的准零刚度局域振子, 其示意图如图7(b)-I所示. 将惯性放大准零刚度局域振子周期性地安装在薄梁上形成超材料梁, 开启的弹性波带隙结构如图7(b)-II所示. 显然, 引入具有小质量块的惯性放大机构能够明显降低弹性波带隙的频率[95].
尽管由惯性放大机构构成超材料或者在局域振子中引入惯性放大机构能够降低和调节弹性波带隙频率, 但是大部分惯性放大机构仍属于被动机械结构, 不便于对低频带隙进行主动或半主动调控. 目前, 通过主动/半主动方式改变局域振子质量以调控带隙的研究, 主要集中在如何快速在基体和局域共振子中转移部分质量. 而电磁铁和泵是两种实现质量转移的主要方式. 详细来讲, 当有电流通过电磁铁时, 连接基体板的两块电磁铁相互粘连. 当关闭电源后, 电磁铁中没有电流通过, 相互粘连的电磁铁脱开, 达到改变局域振子质量的目的[143]. 将超材料基体和局域振子设计成空腔, 并在空腔内填充部分液体. 使用泵实现液体在局域振子空腔和基体空腔之间的切换, 也可以实现局域振子质量的半主动调控[143].
通过主动/半主动或者被动的方式, 调节超材料局域振子的刚度, 也能够对低频弹性波带隙进行调控. 目前, 通过被动方式调节局域振子刚度的途径主要包括在局域振子中引入负刚度机构以抵消局域振子弹性元件的刚度值[144]、镂空超材料基体以减小局域振子刚度[84]以及调节局域振子结构的压缩量[103]等. 研究结果显示, 通过被动调节局域振子刚度的方式能够降低超材料的弹性波带隙频率, 有利于在低频范围内开启带隙结构. 但是, 当被动局域振子设计完成后, 其结构难以更改, 局域振子刚度难以调节, 不便于超材料低频带隙的快速调控.
通过调节局域振子刚度的方式调控超材料低频带隙的第二种方法为构造主动/半主动型超材料. 几种典型的主动/半主动超材料结构示意图如图8所示. 按照控制变量对主动/半主动型超材料进行分类, 可以分为电控类超材料[14, 145-150]、气压控制类超材料[151-152]以及温控类超材料[153-156]. 常见的电控变量主要包括电压、电流、电阻以及电感. 通常, 压电材料、电磁机构以及磁流变弹性体是超材料执行电控变量以控制局域振子刚度的主要执行元器件. 详细来讲, 当在压电材料或磁流变弹性体上作用电场后, 压电材料因逆压电效应产生变形, 而磁流变弹性体的弹性模量发生变化, 实现对超材料带隙结构的调控[14, 157-158]. 而对于电磁机构而言, 其基本原理为通过电磁感应原理将电能转化为机械能, 并辅以合适的机械结构实现对局域振子刚度的调节[130]. 需要指出的是, 如图8(a)所示的压电材料也是构造智能超材料的主要元器件, 其力学性能可以通过调节电控变量进行显著调节.
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如图8(b)所示, 当使用气压作为控制变量对超材料的低频弹性波带隙进行调控时, 其执行元件为气囊. 当气囊内气压增加时, 气囊刚度变大, 局域振子固有频率增加, 超材料带隙向高频范围移动. 相反, 当气囊内气压减小时, 气囊刚度变小, 超材料弹性波带隙向低频范围移动[152]. 温控超材料的执行元器件主要为形状记忆合金, 其示意图如图8(c)所示. 通过形状记忆合金调控超材料低频弹性波带隙的基本原理为当加载在形状记忆合金上的温度发生变化时, 形状记忆合金的组织发生马氏体相与奥氏体相之间的相互转化. 此时, 形状记忆合金的弹性模量可以表示为[153]
$$ E\left( \zeta \right) = {E_{\text{A}}} + \zeta \left( {{E_{\text{M}}} - {E_{\text{A}}}} \right) $$ (10) 式中,
$ \zeta $ 表示形状记忆合金中马氏体所占比例, 其值与马氏体相变开始温度、马氏体相变截止温度、奥氏体相变开始温度、奥氏体相变截止温度以及加载在形状记忆合金上的温度等参数相关; EA与EM分别表示形状记忆合金完全由奥氏体以及完全由马氏体组成时的弹性模量. 显然, 当加载在形状记忆合金上的温度发生变化时, 形状记忆合金的弹性模量发生变化, 导致由形状记忆合金构成的局域振子的刚度发生变化, 进而对超材料的低频弹性波进行调控.3. 低频带隙超材料的潜在应用
当弹性波在超材料结构中传播时, 弹性波与基体/散射体或局域共振单元相互作用, 超材料开启弹性波带隙, 处于带隙范围内的弹性波传播受到抑制[159]. 目前, 超材料已经被用于输流管道的流致振动抑制[160-161]、舰船动力设备与辅助设备产生的低频机械减振[93, 162]、太阳能电池板低频减振[163]、汽车低频减振[164-165]以及蜂巢类轻质结构低频抑振[166]等. 研究结果显示, 工程结构中振动传播的幅值衰减超过20 dB, 有效减小了振动对工程结构的影响.
局域振子的反共振效应是局域共振型超材料产生带隙的本质原因. 当作用在超材料上的弹性波的频率处在带隙内, 即外激励频率接近局域振子的固有频率时, 局域振子动力学响应较大, 超材料能够对机械能量进行俘获[167]. 使用超材料俘获振动能量的两种典型结构如图9(b)所示. 目前, 摩擦纳米发电机[168]、电磁发电机[169]以及压电材料[86]是同超材料相结合以俘获振动能量的主要能量转化机构. 研究结果显示, 将超材料与俘能材料/结构相结合以构成俘能超材料, 能够俘获频率在100 Hz附近的振动能量, 且输出电压可以达到10 V以上.
超材料除可以用于振动抑制和振动能量俘获以外, 还可以执行逻辑运算以实现信息处理[83]. 一般而言, 力学超材料中具有两种不同几何形状的元胞可以用来表示二进制状态0和1. 改变元胞间的相互作用, 力学超材料可以实现不同的逻辑门和信号传输[170]. 目前, 可用于构造力学逻辑超材料的元胞形式包括螺旋弹簧[83]、双稳态弹性结构[171]、导电聚合物[172]以及屈曲梁[170]等. 无论构成逻辑超材料的结构形式如何, 已经提出的超材料都能够实现基本的逻辑运算, 并为实现软体机器人的相关机械逻辑奠定了理论基础. 需要指出的是, 超材料还有诸如构造波导器件[173]、实现声学黑洞[174]等众多应用.
4. 结论与展望
超材料的提出, 引发了包括信息技术、国防工业、新能源技术以及细微加工等诸多领域的重大变革. 带隙是超材料的关键特征之一, 被广泛用于弹性波操控与利用. 超材料带隙结构与超材料的晶格常数、基体与散射体材料属性以及局域振子属性相关, 如何利用小尺度超材料在低频范围内开启带隙结构是扩大超材料在低频弹性波操控与利用领域的关键科学问题之一. 本文在简要介绍超材料的带隙开启机理与影响因素后, 详细总结了能够在100 Hz附近频率区域开启带隙的布拉格散射型超材料、传统局域共振型超材料以及准零刚度超材料, 并厘清了低频弹性波带隙频率与超材料参数之间的关系. 低频弹性波带隙的优化策略和调控方法也在本文中进行了详细阐述. 最后, 从低频振动控制方法、低频振动能量俘获以及逻辑运算等三个方面分析了超材料的潜在应用. 随着对超材料研究的深入, 低频超材料的研究将逐步完善, 并在多个领域实现工程应用.
低频弹性波超材料是近十年来关于超材料研究的新方向, 目前已经在结构设计、数学模型建立以及潜在应用等方面取得了丰硕的研究成果. 但是, 低频超材料距离工程应用还存在诸如承载能力小、结构不够紧凑、非线性模型难以建立与求解等多方面的不足. 因此, 低频超材料的发展还面临如下挑战.
(1)结构化低刚度且具有承载能力的局域振子的创新设计. 传统局域振子存在承载能力与低刚度之间的矛盾, 难以开启低频带隙. 准零刚度超材料为设计具有承载能力的低频带隙超材料提供了契机. 但是, 目前大部分准零刚度局域振子由机械结构组成, 结构复杂、尺寸较大、自重较重, 难以构造紧凑轻质的超材料. 因此, 基于拓扑优化方法, 创新设计具有承载能力的结构化小刚度局域振子, 是低频超材料进一步发展的关键.
(2)非线性超材料的数学建模与分析方法. 传统局域振子由线性/准线性弹性元件提供回复力, 其回复力−位移关系表达式较为简单, 可使用传递矩阵法和平面波展开方法等推导超材料的色散关系, 获得其带隙结构. 但是, 真实局域振子中存在诸如间隙、碰撞、干摩擦、材料弹塑性、结构大变形、几何非线性等非线性因素. 如何对含有非线性的超材料进行数学建模, 并求解超材料的高维非线性方程, 是设计超材料并评估其性能的核心问题.
(3)低频弹性波带隙结构的优化与主动调控. 对局域共振带隙而言, 其带隙宽度随带隙频率的下降而变窄, 带隙内超材料对弹性波的衰减效果减弱. 此外, 目前对低频弹性波带隙的主动调控方法较少, 难以在低频范围内对弹性波带隙进行大范围的主动调控. 因此, 拓宽低频带隙宽度, 强化带隙内弹性波衰减效果以及对低频带隙位置进行主动调控是拓展超材料在振动抑制方面应用的另一关键性问题.
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