-
摘要: 力学超材料是一种有别于传统力学行为的工程材料, 它源于人工可编程的微结构以及材料的固有属性. 在过去的十年中, 随着计算性能和复杂微观结构制造能力的巨大进步, 力学超材料已经吸引了研究人员的广泛关注, 因为它能够实现自然界中不可能出现的多重物理属性. 该领域迅速崛起的趋势之一是将材料行为和单元结构与其他不同的多种物理因素(如电场或磁场)以及温度、光或化学反应等刺激相结合, 从而扩大按需主动调制力学响应的范围. 在本文中, 我们旨在概述有关超材料的力学和多物理性质调制的相关文献, 重点介绍双能级设计的新兴趋势, 着重讨论力学超材料在关键工程领域应用中的巨大潜力. 本文对该领域的发展趋势、挑战和未来路线进行了系统深入分析, 涵盖实时可重构性和功能编程、4D打印、纳米超材料、人工智能和机器学习、多物理折纸/剪纸、活性物质、软物质和保形超材料、复杂微结构制造、服役寿命效应和可扩展性等概念.Abstract: Mechanical metamaterials are engineered materials with unconventional mechanical behavior that originates from artificially programmed microstructures along with intrinsic material properties. With tremendous advancement in computational and manufacturing capabilities to realize complex microstructures over the last decade, the field of mechanical metamaterials has been attracting wide attention due to immense possibilities of achieving unprecedented multi-physical properties which are not attainable in naturally-occurring materials. One of the rapidly emerging trends in this field is to couple the mechanics of material behavior and the unit cell architecture with different other multi-physical aspects such as electrical or magnetic fields, and stimuli like temperature, light or chemical reactions to expand the scope of actively programming on-demand mechanical responses. In this article, we aim to abridge outcomes of the relevant literature concerning mechanical and multi-physical property modulation of metamaterials focusing on the emerging trend of bi-level design, and subsequently highlight the broad-spectrum potential of mechanical metamaterials in their critical engineering applications. The evolving trends, challenges and future roadmaps have been critically analyzed here involving the notions of real-time reconfigurability and functionality programming, 4D printing, nano-scale metamaterials, artificial intelligence and machine learning, multi-physical origami/kirigami, living matter, soft and conformal metamaterials, manufacturing complex microstructures, service-life effects and scalability.
-
1. 引 言
近年来新材料的发展主要集中在调控材料成分, 从而生成各种不同的功能属性. 这种新材料包含所需的性能, 经过设计可以满足特定的应用要求(Fleck et al. 2010). 传统的天然材料具有其自身的特定属性, 固有的力学性质和物理性质是新材料发展的制约因素(Jones & Ashby 2011). 通常的做法是根据可用的材料属性来设计力学和结构, 然而这些传统材料在各种物理性质方面的局限性, 限制了其后续的设计过程. 在多功能系统中, 这些局限性变得更加明显, 特别是需要同时达到多个目标, 或者需要远远超出天然材料极限的极端性能. 力学超材料, 这种新兴概念的材料, 可以在很大程度上解决这些问题, 推动在各个尺度上实现超凡性能, 超越现代结构多功能系统设计的边界. 超材料是一种人工工程材料, 具有周期性(或准周期性, 非周期性)的微观结构或纳米结构, 由“量身定制”(特定应用, 基于功能需求)的几何形状和排列模式组成, 可以产生传统材料无法实现的、前所未有的、非比寻常的宏观特性. 由于它们能够表现出卓越的声学、光学、电磁或力学特性, 近年来, 这些新型结构材料引起了科学界的广泛关注(Chen et al. 2023, Valipour et al. 2022, Zadpoor 2016).
“超材料”一词源于希腊语, 意思是超越物质, 说明了它的超越性. 根据传统材料的经典定义, 超材料可能不符合“材料”的标准. 超材料通常具有(用户)定义的微观结构, 其尺度远低于块体材料性能的尺度. 因此, 尽管它是一种微观或纳米尺度的结构, 但从宏观角度来看, 它是一种材料. 这些都是人为创造的结果, 展示了在天然材料中罕见或无法观察到的物理特性. 它们的构成成分具有明显的空间变化. 人工编程材料可以在纳米层次进行设计, 然后在宏观上表现出超出预期的性能(Pendry 2000, Shelby et al. 2001, Smith et al. 2004)(参考图1(b)). 超材料的非传统特性不仅仅来自于其固有的材料属性, 还来自于其微观结构的几何特性(参考图1(d) ~ 图1(f)). 因此, 在超材料中, 材料和物理性质可以在两个不同的尺度上定义: 一是在较低尺度上, 对应于构成单胞的材料(称为固有材料属性); 二是在宏观尺度上展现的有效特性, 这种宏观特性是固有材料属性和微观结构的综合效应. 超材料通常采用周期性的微观结构形式, 根据功能需求设计单胞, 单胞可以在一维、二维或三维空间堆叠, 形成材料微观结构(Adhikari et al. 2021, Banerjee et al. 2017, Chaurha et al. 2022, Hahn et al. 2020, Kadic et al. 2019, Sinha & Mukhopadhyay 2023). 然而, 根据不同的功能需求, 超材料也可以具有渐变、准周期或非周期的微观结构(Mukhopadhyay et al. 2019c, Mukhopadhyay & Adhikari 2016a, Mukhopadhyay & Adhikari 2017a, Reid et al. 2019, Yao et al. 2021).
在自然界中, 多孔材料表现出高刚度和低密度(Libonati & Buehler 2017), 这是一种力学性能互斥的有益组合. 但这只是一个例外, 成为开发大量力学超材料的灵感来源(参见图1(b)). 很难找到一种天然材料, 能同时满足高刚度和低密度的特性. 此外, 天然材料的性能仅能在有限的范围内得到提高. 一般来说, 天然材料的物理性质, 如密度, 是与它们的力学性质相关的. 例如, 具有高强度和高刚度的传统材料具有高密度, 反之亦然. 在许多技术要求很高的领域, 如航空航天、机器人和机械工程, 都迫切需要打破这种相关性, 开发既坚固坚硬, 同时又很轻的材料. 为了在宏观层面上获得所需的比刚度和比强度, 研究人员在微观层面上投入了大量的努力(参见图1(a) ~ 图1(b)). 在结构设计中, 材料尺度的减小实际上增强了材料的力学特性(Gao et al. 2003, Jang & Greer 2010). 具备小尺度设计的能力, 可在更大的空间中实现具有微观结构可调控的新材料, 从而允许基于特定的应用需求, 获得相应的力学性能(Lee et al. 2012, Moruzzi et al. 2021). 近年来, 3D打印技术的进步使得不同材料复杂微观结构的制备成为了可能(Bauer et al. 2016, Vyatskikh et al. 2018).
正如前文所述, 多孔材料的结构和材料响应依赖于微观结构的几何形状而不是其化学成分. 这些在自然界(骨组织)和人工系统中(桁架桥梁)都大量可见, 涵盖不同的尺度(参见图1(b)). 在自然界中, 像木材和骨头这样的材料具有很高的比刚度, 它们的内部结构使得这些轻质材料能在不失效的情况下承载重物. 单胞结构力学的历史可以追溯到在自然系统中观察到的细胞层次结构(Gibson 2012). 最近的研究工作包括对随机聚合物泡沫的研究, 其中Lakes (1987)对其单胞结构进行了改变, 使得材料具有负泊松比. Gibson等人(Gibson & Ashby 1997, Gibson et al. 1997)的研究工作阐述了作为梁网络的胞状固体的力学性质, 通过调整梁元素的几何参数, 展现出可调控的宏观性质. 图1(c)作为一个典型例子, 展示了正、负和零泊松比的六边形晶格微结构. 这种单胞设计使得结构材料(Ashby 2005)相比于具有更好几何连通性的泡沫材料(Deshpande et al. 2001)拥有了更大的刚度. 几十年后, 随着增材制造技术的进步, 这些复杂结构物理形式的研究在科学界得到了高速推进, 受到了越来越多研究人员的重点关注(Schaedler et al. 2011). 该领域的新兴趋势涉及多物理超材料, 其微观结构不仅能够具备高比强度和高比刚度, 还能够实现多重物理性能, 例如在外部刺激(例如电场和磁场)下主动调制特性, 实现振动和波动控制和能量收集等. 力学超材料领域的最新发展涵盖了不同种类的微结构(见图1(c) ~ 图1(f)), 及其在多功能结构设计中的应用, 因此对这些发展进行综述并对当前趋势和未来路线进行深度分析显得尤为重要.
本文的主要目的是对人工力学超材料的最新发展进行全面系统的分析, 重点关注主动性、可重构和多物理行为. 在本文中, 我们首先介绍基于功能和微观结构的新兴超材料类别, 然后我们将重点转移到具有双级被动和主动行为的力学超材料上. 进一步讨论不断发展的多功能性概念, 其中微结构的力学行为与超材料的主动微元组分相结合, 共同实现多物理场行为. 随后, 我们将重点转移到通过制造设计的复杂材料微结构来实现物理性能的相关问题. 最后, 我们总结了科学界亟待解决的关键问题, 以及力学超材料领域的长期路线和发展前景.
2. 基于功能和微观结构的新兴超材料分类
超材料可以根据功能和微观结构构型进行分类(参见图2). 在本节中, 我们简要讨论不同的新兴超材料种类. 值得注意的是, 超材料的新趋势包含多功能性, 其中超材料可用于实现涵盖多重物理场的设计目标.
2.1 基于功能的分类
超材料的设计目标是基于其意图输出特定的特性, 即所期望的功能性. 超材料是在较低的尺度(如微米或纳米尺度)上设计的, 以便在较大的尺度(如宏观尺度)上实现这些功能. 根据均质化理论得到的有效参数(Bakhvalov & Panasenko 1989, Jikov et al. 1994), 可以将超材料大致分为四类, 即电磁超材料、光学超材料、声学超材料和力学超材料. 在深入讨论本文的重点−力学超材料之前, 我们先简要介绍这些分类.
2.1.1 电磁超材料
电磁超材料是通过调控微观参数而获得的人工设计材料, 其结果是对光与物质的相互作用实现特殊调控. 这些超材料对光的电、磁分量产生不同的响应 (参见图3(a)). 电磁超材料具有不寻常的特性, 例如负磁导率和负介电常数. 负磁导率意味着外加磁场作用下, 在与施加磁场相反的方向上形成磁偶极子. 类似地, 负介电常数意味着电场矢量和电位移矢量指向彼此相反的方向. 这种超材料有多种应用, 例如天线(Lier et al. 2011, Lier & Shaw 2008)、隐形斗篷(Cai et al. 2007, Schurig et al. 2006)、“完美透镜”(Pendry 2000, Podolskiy et al. 2005)和不同类型的传感器(Alù & Engheta 2008, Jakšić et al. 2007, Lee & Yook 2008).
2.1.2 光学超材料
光学超材料(Chen Y et al. 2020, Shalaev et al. 2005, Soukoulis & Wegener 2010)是指人工设计的微观结构具有独特的光学功能, 如负折射(Shalaev 2007, Shelby et al. 2001, Valentine et al. 2008)、双曲色散(Kruk et al. 2016, Poddubny et al. 2013)和超分辨率成像(Fang et al. 2005, Jacob et al. 2006, Zhang & Liu 2008), 这在自然材料中是不常见的. 负折射率意味着当光线从一种介质传播到另一种介质时, 入射与折射在界面法线的同一侧(参见图3(b)(i) ~ 图3(b)(ii)). 这需要在局部水平上对穿过材料的电磁波阻抗和相速度等特性进行调控(参见图3(b)(iii) ~ 图3(b)(vi)). 早期的光学超材料是超表面或超薄膜. 然而, 目前已经设计获得了各种各样的3D光学超材料结构(参见图3(b)(vii)). 同样, 新近研发的近场型的超级透镜, 它的成像分辨率不受波长的限制, 而仅受材料质量的影响.
2.1.3 声学超材料
可以使用声学超材料在亚波长尺度上控制和操纵声波. 变换声学和高度各向异性的声学超材料的耦合机制, 可以调控声场的传播(Cummer et al. 2016). 声学超材料可以具有负质量密度(Lee et al. 2009a)或负体积模量(Lee et al. 2009b). 当有效密度和体积模量同时为负时, 可获得双负声学系统(Li & Chan 2004). 需要注意的是, 这些独特的属性在天然材料中较为罕见. 负质量密度意味着对于负质量, 加速度方向相反. 负体积模量则意味着体积随着压力的施加而增加. 这些前所未有的属性可用于声学超级透镜(Ambati et al. 2007, Guenneau et al. 2007)、隐身(Cummer & Schurig 2007)和小型超声成像元件(Zhang et al. 2009). 这个领域的最新研究包括柔性3D声学超材料(Brunet et al. 2015), 它可以用于对声学参数要求为负数的亚波长成像和声学变换等器件 (参见图3(c)) .
2.1.4 力学超材料
受到超材料家族前辈的启发, 在过去十年左右的时间里, 一种新型的超材料逐渐崭露头角, 致力于提升材料的有效力学性能. 起初, 力学超材料领域仅限于实现一些熟悉物理属性的非常规值 (零或负), 例如泊松比(Huang & Chen 2016, Lakes 1993, Lakes 1987)、密度(Oh et al. 2016)或可压缩性(Imre 2014). 然而, 近年的研究提出了相对较新的力学超材料, 包括拓扑优化(Kozin et al. 2018, Krishnamoorthy et al. 2012)、可变构型(Dudek et al. 2022, Jiang et al. 2022, Zhang & Krushynska 2022)和非线性超材料(O’Brien et al. 2015, Shadrivov 2010). 这些材料显示出超凡的功能, 例如可编程刚度(Florijn et al. 2014)、多变的形状和图案(Chen & Jin 2018)、耗散(Bacquet et al. 2018, Lee et al. 2020, Ma et al. 2022)或运动和波的单向引导(Nash et al. 2015, Tian & Shen 2020). 力学超材料可以具有负杨氏模量和负泊松比. 负杨氏模量意味着单元的挠度与所施加载荷的方向相反(Adhikari et al. 2020). 当对具有负泊松比的材料施加纵向压缩时, 它们会横向收缩, 反之亦然(Mukhopadhyay & Kundu 2022, Sinha & Mukhopadhyay 2022, Wang H et al. 2020)(参见图3(d)). 本文稍后将详细介绍不同被动和主动力学超材料的最新进展及其多物理场行为.
由于这篇综述论文的重点是力学超材料, 我们在图4 中进一步展示了它们当前和潜在的应用前景. 请注意, 这里讨论的应用只是此类超材料无数工程示例中的一小部分. 例如点阵结构可以用作隐形斗篷, 以逃避对物体的主动探测. 这种斗篷可在国防领域中得到重要的应用 (参见图4(a)). 3D打印的进步推动了柔韧性可调复杂微观结构的制造, 从而产生了用于软体机器人、电子皮肤、仿生夹具、柔性电池等的超材料(参见图4(b) ~ 图4(d)和图4(i)). 力学超材料的最新应用还包括机械计算和逻辑门(见图4(e)). 同时, 拉胀力学超材料是一种优异的抗冲击材料 (见图4(f)). 最近发现, 拉胀力学超材料还可用于鞋类行业(见图4(h)). 拉胀力学超材料作为安全带材料使用时, 其拉胀特性使得汽车行驶更加安全. 安全带在纵向延伸的情况下横向扩展, 并覆盖人体胸部的更大部分, 这在传统材料中是不可能实现的(见图4(k)). 类似的应用还见于绷带, 当伤口沿一个方向扩张时, 可以覆盖更大的伤口创面, 从而帮助愈合伤口 (见图4(m)). 拉胀晶格在桁架核心翼型中显示出优异性能(参见图4(j)). 此外, 蜂窝夹芯结构广泛应用于航空航天等轻量化结构领域(Mukhopadhyay & Adhikari 2016b). 具有适当单胞形状的蜂窝结构在海上建筑物防护方面具有巨大的应用潜力(参见图4(l)). 在医疗应用领域, 力学超材料可用作心血管支架, 通过维持适当的血液流动来帮助保持患者心脏健康(见图4(g)). 这里讨论的应用, 本质上还都是被动的. 随着主动力学超材料的出现, 它将有可能按照功能来调节形状和力学性能, 为未来的工程应用又增加了一个全新的维度, 显著提高性能.
2.2 微观构型
基于微观结构, 超材料主要分为基于晶格结构的超材料和基于折纸/剪纸的超材料两大类. 大部分微观结构具有周期性, 但是它们也可以具有分形、准周期和非周期性质(Mukhopadhyay et al. 2019, Mukhopadhyay & Adhikari 2017a, Reid et al. 2019, Yao et al. 2021). 微结构的周期性使得晶胞的材料属性可以在空间内变化, 或者在周期性微结构中存在多种材料 (称为多材料晶格)(Mukhopadhyay et al. 2020c, Pahlavani et al. 2022). 在本节中, 我们简要描述不同类别超材料的微观结构, 重点陈述其新兴趋势.
2.2.1 基于晶格的超材料
根据晶格的构成元素, 晶格可以分为: 基于板的晶格、基于支柱/梁的晶格和基于三周期最小表面 (TPMS) 的晶格 (有关不同类型晶格的更多详细信息, 请参阅(Zhong et al. 2023)) . 基于晶格的超材料可以分为两大类, 即不同形状的晶格和受拉伸/弯曲主导的晶格.
2.2.1.1 不同形状的晶格
平面晶格是通过多边形堆叠从而填充整个平面获得的. 传统上, 典型多边形蜂窝结构可以是规则的、半规则的或不规则的(Fleck et al. 2010). 规则蜂窝由单个周期性重复的正多边形形成, 或者由六边形、三角形和正方形三种传统形状的晶格形成 (参见图5(a)). 半规则晶格是通过两个或多个不同规则形状的堆叠获得的(Cundy & Rollett 1961, Lockwood & MacMillan 1978). 像Voronoi晶格(Lorna J Gibson & Michael F Ashby 1999)或Penrose (de Bruijn 1981) 这样的不规则晶格是通过重复两个或多个不同大小的不规则多边形形成的. 虽然大多数提出的超材料通常遵循周期结构, 由于引入更多局部空间特征的范围, 非周期晶格正在迅速进入力学超材料的设计领域(Huang et al. 2023, Isanaka et al. 2022).
除了这个分类之外, 晶格基材料可分为手性蜂窝和反手性蜂窝. 手性蜂窝表现出一些非常规的特性, 例如负泊松比(Lakes 1987, Prall & Lakes 1997)或负动态体积模量(Liu et al. 2011). 手性结构是那些不能叠加在其镜像上的结构(Thomson & William 2010, Wu et al. 2017). 手性结构由恒定半径的圆形元件组成, 这些元件充当节点, 通过切向附着的纽带相互连接. 通过改变纽带和节点的空间排列, 可以获得不同类型的手性晶格. 例如, 通过将连接每个节点的纽带数目分别更改为3、4和6, 可以从经典的六边形、矩形和三角形晶格衍生出三手性、四手性和六手性晶格. 此外, 通过纽带同侧连接相邻节点可以获得反手性晶格. 同样基于连接元件的数量, 即3、4和6, 这些可以分别是反三手性、反四手性和反六手性 (参见图5(b)). 其他新型的周期性晶格几何形状包括分形超材料和分级超材料(Mousanezhad et al. 2016, Zhang et al. 2021).
2.2.1.2 以拉伸和弯曲为主的晶格
以弯曲为主的多孔材料 (如泡沫) 主要是依赖其元件 (如梁和板) 的弯曲而变形, 而以拉伸为主的结构则是通过其互连构件的单轴拉伸或压缩而变形. 拉伸主导的结构通常由互连构件的三角形排列表示 (参见图5(c)). 弯曲和拉伸的主导作用可以通过它们的拓扑构型来区分. 研究发现, 拉伸主导的结构其比强度和比刚度高于弯曲主导的结构(Ashby 2005), 而弯曲主导的晶格通常表现出更多的能量吸收能力. Gibson 和 Ashby (Fleck et al. 2010, Lorna J Gibson & Michael F Ashby 1999) 提出缩放律, 应用标准梁理论来研究晶格材料的力学行为, 表明变形机制取决于拓扑结构以及材料分布. 弯曲主导和拉伸主导结构的相对刚度
$ E/{E}_{{\mathrm{s}}} $ 和相对强度$ \sigma /{\sigma }_{{\mathrm{s}}} $ 与相对密度$ \rho /{\rho }_{{\mathrm{s}}} $ 的关系如下$$ 伸展为主: \left(E/{E}_{{\mathrm{s}}}\right)\propto \left(\rho /{\rho }_{{\mathrm{s}}}\right),\left(\sigma /{\sigma }_{{\mathrm{s}}}\right)\propto \left(\rho /{\rho }_{{\mathrm{s}}}\right) $$ (1) $$ 弯曲为主: \left(E/{E}_{{\mathrm{s}}}\right)\propto {\left(\rho /{\rho }_{{\mathrm{s}}}\right)}^{3},\left(\sigma /{\sigma }_{{\mathrm{s}}}\right)\propto {\left(\rho /{\rho }_{{\mathrm{s}}}\right)}^{2} $$ (2) 其中, 这里的相对刚度
$ E/{E}_{{\mathrm{s}}} $ 是晶格的有效杨氏模量$ E $ 与构成材料的杨氏模量$ {E}_{{\mathrm{s}}} $ 的比值. 类似地, 相对强度$ \sigma /{\sigma }_{{\mathrm{s}}} $ 是晶格的有效强度$ \sigma $ 与构成材料的强度$ {\sigma }_{{\mathrm{s}}} $ 的比率. 相对密度$ \rho /{\rho }_{{\mathrm{s}}} $ 是晶格密度$ \rho $ 与组成固体材料的密度$ {\rho }_{s} $ 之比. 由于晶格材料的相对密度$ \rho /{\rho }_{s} $ 通常小于0.2 (Lorna J Gibson & Michael F Ashby 1999), 我们可以得出结论, 拉伸主导的结构比弯曲主导的结构更强、刚度更大.2.2.2 基于折纸/剪纸的超材料
折纸源自日语单词“ori” (意为“折叠”) 和“kami” (意为“纸”), 这是一种近来引起科学界广泛关注的折纸艺术. 由于其密实堆装的能力, 几十年来, 折纸的概念已广泛应用于可展开的空间结构中. 最近, 随着制造能力的进步, 折纸技术正受到越来越多的关注, 特别是在开发具有优异力学特性的材料微结构领域. 例如可编程刚度、负泊松比和多向拉胀、静态变形、极端变形和多稳定性(Che et al. 2016, Evans et al. 2015, Fang et al. 2018, Fang et al. 2016, Schenk & Guest 2013, Waitukaitis et al. 2015). 它们主要遵循板铰链机制, 其中板通过柔性铰链连接, 从而产生具有可调变形机制的复杂几何形状. 这种以折纸为基础的超材料在不同的变形系统、医疗支架、电子设备、DNA纳米制造和力学零件中都有潜在的应用 (如杠杆、滑轮、齿轮等)(Al-Mulla & Buehler 2015, Bertoldi et al. 2017, Kuribayashi et al. 2006, Ning et al. 2018, Rogers et al. 2016, Rothemund 2006, Song et al. 2014, Yu et al. 2018). 基于折纸的超材料可以利用折叠和组装2D平面材料的原理产生复杂的3D结构 (参见图5(d) ~ 图5(e)). 这种超材料的几何形状本质上取决于折痕和顶点两个参数, 最终的形状基于折叠的顺序、方向、大小和数量(Demaine & O’Rourke 2007, Lang 2012).
除了基于折纸的超材料之外, 还可以运用剪纸技术 (切割纸张) 来制备超材料. 使用这种技术可以调控材料的弹性特性, 以获得极端应变或拓扑变化(Cho et al. 2014, Gatt et al. 2015, Isobe & Okumura 2016). 通过剪纸技术制作的图案化片材可表现出屈曲和超平面形变, 从而在多个尺度上形成复杂的三维结构, 这使其在结构和力学应用方面具有广泛的适用性(Lamoureux et al. 2015, Liu et al. 2018, Rafsanjani & Bertoldi 2017, Xu et al. 2016, Zhang et al. 2015). 此外, 剪纸的设计原理可以和基于折纸的材料相结合, 形成具有不同切割图案的复杂折叠结构(Eidini & Paulino 2015, Sussman et al. 2015). 这从根本上扩大了基于特定机制的超材料设计空间.
基于折纸和剪纸的超材料可大致分为基于折纸的超材料、基于折纸/剪纸的混合超材料和基于剪纸的超材料. 这些超材料可以是刚性的, 也可以是可变形的 (参见图5(f)(i) ~ 图5(f)(vi)). 折纸超材料可以根据不同的模式继续细分为如Miura (Dudte et al. 2016)、组装Miura (Filipov et al. 2015)、waterbomb (Mukhopadhyay et al. 2020)、eggbox (Fathers et al. 2015, Pratapa et al. 2018)、kresling (Zhai et al. 2018)、square-twist (Silverberg et al. 2015)以及其他不同的多自由度刚性折纸(Lee et al. 2014). 刚性混合折纸/剪纸可以被切割和折叠(Tang et al. 2019)并组装以获得新的架构(Overvelde et al. 2016). 此外, 刚性剪纸可以细分为2D (Choi et al. 2019)和3D刚性剪纸(Jin et al. 2020). 根据折痕的性质, 折纸可以分为直折痕(Silverberg et al. 2014)和曲线折痕(Zhai et al. 2020). 可变形剪纸可分为拉伸屈曲(Blees et al. 2015)、压缩屈曲(Zhang et al. 2015)和自变形剪纸(Liu et al. 2018)(参见图5(f)(vii)). 可变形混合折纸/剪纸超材料(Novelino et al. 2020, van Manen et al. 2020)可以根据所采用的折纸或剪纸图案的性质进行分类.
3. 力学超材料的多物理场特性调制
力学超材料是一类通过排列不同的微观结构以在宏观尺度上实现特定力学性能而设计获得的超材料(Frenzel et al. 2017). 这些人工设计的微观结构其有效特性不仅取决于组成元素的固有特征, 还取决于几何和结构构型(Meeussen et al. 2016, Mukhopadhyay & Adhikari 2017b, Zheng et al. 2014). 这种超材料可以通过微结构的智能设计提供各种前所未有的、非常规的、极致和有益的特性(Chen et al. 2014, de Moura et al. 2022, Nash et al. 2015, Paulose et al. 2015, Wang H et al. 2020), 例如负刚度(Tan et al. 2020)、趋于零的剪切模量(Bückmann et al. 2014, Kadic et al. 2012)、负压缩率(Gatt & Grima 2008, Lakes et al. 2001, Nicolaou & Motter 2012)、负泊松比(Evans & Alderson 2000, Lakes 1987, Lim 2015, Milton 2015, Prall & Lakes 1997)和奇异非线性行为(Gómez et al. 2012, Wyart et al. 2008). 与传统复合材料 (例如纤维增强层压板)(Chandra et al. 2022, Dey et al. 2018, Mukhopadhyay et al. 2021, Vaishali et al. 2023)相比, 力学超材料从根本上提供了可扩展的设计空间, 并显著改善了多功能属性调制的范围(dell’Isola et al. 2016). 因此, 力学超材料使用拓扑优化 (结构化或非结构化) 而不是材料成分来获得最终的性能(Del Vescovo & Giorgio 2014).
对单胞层面的拓扑结构进行深入研究可以得到不同的力学特性, 在纯粹依赖单胞几何形状的超材料发展中已经达到了相对饱和的水平. 为了突破瓶颈, 其中一个不断发展的趋势是采用双级设计, 即对单胞的基本构成组件 (例如梁/板状元件) 进行进一步设计(Sinha & Mukhopadhyay 2023, Singh et al. 2022a, Singh et al. 2022b, Singh et al. 2021, Sinha et al. 2023b, Sinha & Mukhopadhyay 2023b). 基本构成单元层面和晶胞层面所涉及的力学相互作用, 通过可扩展的设计空间实现极致特性. 通过这个双级框架, 可以同时考虑梁层和单元层变形的被动物理力学和主动物理力学 (例如梁这一级别固有应力的影响(Sinha & Mukhopadhyay 2022)、固有材料的黏弹性(Mukhopadhyay et al. 2019c)、或多物理行为, 如电(Singh et al. 2022b, 2021)和磁(Singh et al. 2022a)的主动变形). 值得注意的是, 在超材料中引入主动元素 (其中变形可以通过磁场或电场等非力学外部刺激来控制), 可对超材料的实际本构行为进行按需主动调制, 包括弹性模量等基本性质. 在以下小节中, 我们将讨论主动和被动力学超材料的最新进展, 重点是双级设计和多物理机制行为.
3.1 被动特性调制
制造后无法进行特性调制的力学超材料称为被动超材料. 在这种类型的超材料中, 基于微观结构设计, 即可实现预期性能 (包括多功能特征)(Lorna J Gibson & Michael F Ashby 1999)(参见图6). 在本小节中, 我们将讨论材料微观结构发展中的一些最关键和最突出的功能.
3.1.1 有效弹性性能和力学本构行为
被动力学超材料的弹性特性依赖于固有的材料属性和微观结构几何形状. 表征有效弹性特性对于将这些超材料应用在不同的结构和力学应用中至关重要. 蜂窝晶格是力学超材料最典型的构型之一, 因此已对其不同的有效特性进行了广泛的研究. 该领域的研究包括在考虑梁这一级别的弯曲、轴向和剪切变形推导的情况下推导正六边形蜂窝结构的有效弹性特性(Abd El-Sayed et al. 1979, Lorna J Gibson & Michael F Ashby 1999, Zhang & Ashby 1992). 考虑到不同的单元几何形状和相对密度, 对周期性六边形蜂窝 (包括其衍生物, 如菱形、矩形和拉胀构型) 进行了数值和解析分析(Malek & Gibson 2015). 学者研究了缺陷 (如未对准或蜂窝壁破裂、蜂窝尺寸变化、蜂窝壁厚度不均匀和蜂窝缺失) 对蜂窝力学性能的影响(Chen et al. 1999, Wang & McDowell 2003); 以及非周期Voronoi微观结构 (参见图6(e)) 对蜂窝有效弹性性能的影响(Silva et al. 1995); 通过实验和数值研究了不同微观结构几何参数 (例如单元长宽比、相对厚度和单元角度) 对蜂窝结构杨氏模量和泊松比的影响(Scarpa et al. 2000); 等几何分析和模型降阶技术可用于辅助超材料的三维拓扑优化 (Nguyen et al. 2020). 在近期的一些工作中, 有学者使用解析方法获得了不规则蜂窝晶格结构的等效面内弹性特性(Mukhopadhyay & Adhikari 2016c, Zhu et al. 2001). 此外, 还有学者研究了粘弹性效应和结构的不规则性(Mukhopadhyay et al. 2019c). 已从理论和数值角度对类似晶格的二维材料弹性特性开展了纳米尺度的分析, 包括纳米异质结构的力学表征, 这实质上引入了纳米尺度上结构设计的概念 (Chandra et al. 2020; Gupta K et al. 2022; Gupta K K et al. 2022; Mahata & Mukhopadhyay 2018; Mukhopadhyay et al. 2018, 2017a, 2017b, 2020b)(参见图6(h)).
通过双级分析框架(Sinha & Mukhopadhyay 2022)(参见图6(a)), 学者研究了二维蜂窝结构中不可避免的固有应力对结构的影响机理, 这些固有应力可视为一种制造上的结构不规则性. 有学者通过解析方法(Sinha et al. 2023a), 将这种双级分析进一步扩展到三维晶格, 研究了三维晶格的有效弹性性能 (参见图6(b)). 值得注意的是, 这种横梁层级的固有应力也可视为一种增益效应, 有助于改善和调节结构层级的有效性能. 通过在蜂窝细胞壁中引入区域性的不连续性, 在线性变形状态下实现了法向和剪切模式下的可变弹性模量(Sinha et al. 2023b) (参见图6(g)). 在拉伸和压缩或顺时针和逆时针模式下, 有效弹性模量可以根据不连续区域的位置和程度进行编程, 使其具有差异性. 近来引入了晶格反曲率的概念, 并基于双级半解析非线性框架对有效弹性模量和失效强度进行了表征(Ghuku & Mukhopadhyay 2023, Ghuku & Mukhopadhyay 2022, Prajwal et al. 2022)(参见图6(d)). 根据反曲率的程度, 这些晶格的强度和刚度可以显著增强. 有学者提出了对面外曲率编程来增强弯曲主导复合材料晶格的多模态刚度(Tiwari et al. 2023), 同时保持其传统的多功能优势, 例如高能量吸收能力. 极高的比刚度可以通过微拓扑架构的超材料中的双层蜂窝网络来实现(Kundu et al. 2023), 除了晶胞级几何结构之外, 还对构成晶格的横梁状元素进行了进一步地拓扑优化. 最近的一项研究提出了一种受剪纸启发的模块化超材料, 可实现接触诱导的刚度调制和本构关系编程(Sinha & Mukhopadhyay 2022c). 混合模式的多向拉胀性 (包括从非拉胀行为到拉胀行为的转变, 反之亦然) 和可编程刚度进一步表明, 可以通过受折纸和剪纸启发的超材料实现, 而无需外部的非力学刺激 (Mukhopadhyay et al. 2020, Sinha & Mukhopadhyay 2022c, Wang H et al. 2020). 上述有关有效弹性模量和本构行为被动属性调制的文献综述表明, 单胞几何层次的影响已经得到了广泛的研究, 同时探讨了利用内在材料物理属性和双层设计框架的最新趋势. 特别强调的是, 超材料的弹性模量和力学本构行为对于分析结构的静动态特性至关重要.
除了弹性模量和本构行为的讨论之外, 力学超材料还可以进一步分为极值材料和负材料. 极值材料会在某些变形模式下非常坚硬, 而在其他模式下则非常柔顺(Milton & Cherkaev 1995). 极值材料可以进一步细分为五模超材料和膨胀超材料. 与剪切模量相比, 五模超材料(Kadic et al. 2012)具有非常大的体积模量. 高体积模量值导致变形时没有体积变化, 并且非常小的剪切模量值使其类似于流体. 因此, 五模材料也称为“超流体”. 五模超材料将弹性动力波转向所需方向, 从而实现声波的光学隐形. 另一方面, 膨胀材料(Milton 2015)与体积模量相比, 具有非常大的剪切模量. 这意味着在变形时, 只有它们的尺寸发生变化, 而形状保持不变, 从而提高了抵抗变形破坏的能力. 负超材料可以是具有负弹性模量或负体积模量的材料(Lakes & Wojciechowski 2008). 负弹性模量意味着负刚度. 负刚度超材料沿与所施加的外力相反的方向偏转. 类似地, 负体积模量导致负压缩率. 这意味着当施加静水压力时, 这些材料会膨胀. 除了极值超材料和负超材料之外, 还有表现出超凡性能特征的力学超材料, 即合理设计的力学超材料可以表现出超高刚度、高韧性、高强度, 同时可以具有低质量密度(Meza et al. 2014, Zheng et al. 2014). 近来, 研究人员开始关注被动力学超材料的疲劳行为, 重点是通过设计来提高疲劳极限(Benedetti et al. 2021). 上述特性是传统材料所不具备的.
3.1.2 波的传播与振动
波传播和振动是动态环境下力学超材料设计的重要特征(Brillouin 1953, Cai et al. 2021). 自60年代中期以来, 人们就开始研究周期性结构的动力学行为(Mead 1996). 但力学超材料一词在那个时期并不流行. 大多数研究对超材料中波传播的研究都是基于Floquet−Bloch 定理, 这是一种考虑晶胞边界条件的计算方法(Hussein et al. 2014). 晶胞的波传播响应本质上是一个动态特征, 可以使用带隙(Brillouin 1953, Deymier 2013)的概念来理解它. 因此, 最近的研究集中在使用数值和解析方法来评价超材料的带隙(Bigoni et al. 2013, Hussein 2009, Palermo & Marzani 2016, Sugino et al. 2016). 在大量此类文献中, 使用经典的波传播方法对无阻尼超材料进行了研究. 一些研究人员考虑了阻尼超材料中晶胞内部的阻尼(Fan et al. 2017, Hussein & Frazier 2013). 近来大量研究开始采用机器学习来研究波传播行为(He et al. 2021, Jin et al. 2022).
振动是超材料分析和设计的一个重要方面, 因为它可能导致共振和疲劳, 从而导致结构失效. 因此有必要去控制、抑制或减轻不良振动(Inman 2017). 在传输路径中使用波屏障, 可以通过干扰“源-接收器”振动路径来减少不必要的干扰(Jena et al. 2013, Fahy 2018). 振动的被动控制包括将系统的固有频率重新定位到远离工作范围. 这是通过添加阻尼材料层、改变刚度和增加质量来实现的. 这种被动控制不需要任何复杂的系统, 仅通过改变系统的几何参数来吸收振动. 该领域的研究包括使用力学超材料来抑制振动. 它将周期性结构的思想与结构力学性能的控制以及智能材料的使用结合起来. 使用弹性波过滤的声子晶体在周期性结构的基础上工作. 由于波在周期性排列的介质中传播时引起的衍射, 波的色散特性发生变化 (参见图6(f)). 空间晶体排列决定布拉格带隙位置. 振动控制可以通过使用薄超材料板、微结构超材料、准零刚度超材料以及使用带有嵌入式吸收器的超材料梁来实现(Aghighi et al. 2019, Fan et al. 2020, Nouh et al. 2014, Zhu et al. 2011). 有学者提出了复合智能梁与谐振并联电路的被动超材料, 可用于面向阻抗随机失调的波传播和振动控制领域(de Moura et al. 2022, Machado et al. 2022). 这种被动控制技术依赖于基本结构的修改, 而不需要外部刺激源. 近年来研究了动态环境下晶格超材料有效弹性模量的频率依赖性(Adhikari et al. 2021; Mukhopadhyay et al. 2019a, 2019b). 结果表明, 有效弹性模量在较高频率下可以显著增加, 这可用于在振动环境的条件下进行结构优化设计.
3.1.3 能量吸收
常规吸能力学超材料的设计通常引入弯曲主导的微观结构(Chen et al. 2021). 将晶胞设计为具有能量耗散特性的晶胞壁, 然后通过晶胞的周期性重复获得超材料(Zhang J et al. 2020). 2D和3D晶格超材料的能量吸收能力都已被大量研究(Ji et al. 2022, Yang & Ma 2019). 有学者提出一种新型能量吸收力学超材料(Zhu et al. 2021), 它使用具有负刚度的双相材料组件, 将刚度与柔度相结合. 最近的一项研究利用机器学习提出一种基于失效模式驱动的蜂窝晶格材料, 以提升强度和吸能能力(Isanaka et al. 2022). 此外, 折纸图案也可用来设计具有提升能量吸收特性的新型超材料(Xiang et al. 2020).
随着制造能力的进步, 不同形式的吸能结构引起了研究人员的极大关注(Ha et al. 2018). 为了提高性能, 人们提出了一种屈曲诱导的吸能力学超材料(Hu & Burgueño 2015). 大量基于梁结构的吸能超材料(Alturki & Burgueño 2019, Liu S et al. 2019)的研究分析了梁结构各种参数之间的关系及其与能量吸收能力的相关性(Hua et al. 2020, Shan et al. 2015, Zhang Y et al. 2020). 有学者使用双稳态曲梁模型设计了用于能量吸收的多稳态力学超材料(Giri & Mailen 2021). 因此, 通过合理设计晶胞壁及其方向, 可以提高力学超材料的吸能能力.
3.1.4 能量收集
通过控制能量流的大小, 力学超材料可用于能量收集. 例如, 在声反射表面中, 可吸收声波并将其转化为其他形式的能量(Ma et al. 2014, Qi et al. 2016). 研究表明, 受超材料启发的平面结构可以通过波引导、波聚焦和能量局部化来收集机械波能量(Carrara et al. 2013, Carrara et al. 2012). 相对于布拉格散射型声子晶体, 局部共振超材料的谐振腔可以随机分布在材料中(Claeys et al. 2014, Liu et al. 2000). 薄膜型超材料可以用来衰减波, 并通过其中引入的能量收集装置收集能量(Chen et al. 2019). 有学者提出了一种将周期性结构与谐振腔中的能量收集特性相结合的新概念超材料(Krödel et al. 2015, Lv et al. 2013). 具有缺陷的声学压电超材料可以通过引入共振缺陷来捕获声学入射产生的应变能, 从而充当声能收集器(Qi et al. 2016). 最近提出的拓扑表面波超材料, 可以实现稳健的振动衰减和能量收集的双重功能(Wu et al. 2022).
3.2 主动特性调制
在成型之后依然能够按需调制实现有效性能的力学超材料, 是具有主动调制特性的材料. 一般是通过在单胞中使用活性材料, 并通过磁场或电场等外部刺激激活它们来实现 (参见图7). 该领域的最新研究包括将具有刺激响应的材料与 (基于晶胞) 力学超材料微观构型设计的耦合. 在外部刺激下, 这些超材料可以表现出非常规的特性, 因此可以根据现场操作的需求, 按照不同的特定应用要求来使用. 外部刺激可以是压力作用(Chen & Jin 2018, Rafsanjani et al. 2019)、热(Yang et al. 2019, Yang & Ma 2020)、光(Wang L C et al. 2020)、磁场(Gu et al. 2020, Jackson et al. 2018, Montgomery et al. 2021)、电流(El Helou et al. 2021, Zhang Z et al. 2017)和化学物质(Li et al. 2017, Zhou et al. 2018). 此外, 形状记忆合金也可用于超材料中, 以实现包括形状变形在内的主动行为(Yang et al. 2023). 需要注意的是, 这种主动性质调制通常涉及到力学超材料的多物理场行为, 包括在力学负载和非力学刺激下的变形力学. 因此, 这种元梁级、微米级和晶胞级的多物理场行为可以通过双级设计范式在宏观超材料级表现出前所未有的主动特性. 在下面的小节中, 我们将讨论主动超材料在不同外部刺激下展现多物理场行为的最新进展.
3.2.1 热响应主动力学超材料
热驱动方法的工作原理是控制热交换来改变热响应材料的行为. 从广义上讲, 热驱动涉及光热效应、磁热效应、电热效应和热化学反应, 是超材料主动力学性能调节最广泛的驱动方法之一. 直接或通过改变周围环境的温度将热量传递到超材料. 使用最为广泛的典型力学超材料微结构, 即晶格和折纸结构, 已经实现了热响应的主动性能调制. 有学者开发了轻质主动调制晶格, 观察到它的刚度随温度变化而发生显著变化(Yang et al. 2019). 在冲击载荷作用下, 这些结构可以显著吸收冲击力, 即使在发生较大变形的情况下也可以恢复到初始形状. 近来, 4D打印可用来生成拉胀热激活折纸超材料(Xin et al. 2020), 学者探索了其变形行为和力学性能. 还有学者开发了基于手性构型响应外部热刺激的超材料(Lei et al. 2019, Tao et al. 2020). 研究人员开发了一系列在热刺激下从平面状态获得不同形状的简单结构, 如壳、管、扇形、棒等(Ding et al. 2017). 有学者制备了具有鲁棒刚度、可延展性和特定温度下可恢复变形的超材料(Wu et al. 2021)(参见图7(i)). 从1D到3D的多稳态结构设计中, 研究人员实现了热膨胀系数的可控性(Yang & Ma 2020). 这种热膨胀系数可调的力学超材料(Boatti et al. 2017), 其折纸面板的折痕排列在特定的位置. 随着温度的变化, 热膨胀系数可为正值、零值和负值. 使用模块化的4D打印技术可以制造具有不同玻璃化转变温度的结构, 由于温度变化引起的结构变形可以任意配置(Fang et al. 2020). 因此, 我们看到热能已经广泛用作主动材料设计中的外部刺激, 具有按需调节形状和力学性能的能力.
3.2.2 光响应主动力学超材料
光响应材料的力学行为受外界光环境的控制. 通过在聚合物试剂中加入光敏性官能团可以获得对光主动响应的材料性能 (参见图7(f) ~ 图7(g)). 因此, 光驱动可以是光热或者光化学驱动(Zeng et al. 2018). 光热驱动是热刺激的一种, 因此类似的概念也适用于力学超材料的微观结构设计, 而光化学刺激则不受温度的影响. 与光热效应相比, 光化学效应更为复杂. 这些由光驱动的力学超材料综合了光敏感材料的多物理场、力学与微观结构几何形状的设计. 有学者利用光聚合中体积收缩的效应设计了一种可展开的三维折纸结构(Zhao et al. 2017)(见图7(e)), 其中光照时间可以控制结构变形. 此外, 有学者系统设计了一种快速变形且适应性强的4D打印材料(Zhang Q et al. 2020). 当光使聚合物中的挥发性物质挥发时, 结构发生变形. 挥发后, 对聚合物网络中残留的非挥发性物质进行光固化. 研究人员分析了外部光刺激下方形扭曲折纸结构的变形力学行为(Wang L C et al. 2020). 光敏铰链会因光而收缩, 这可用于超材料的设计(Zhang Q et al. 2017). 由于其快速、准确和可控的响应, 这些超材料在软体机器人领域得到广泛应用. 学者基于毛毛虫爬行的想法概念设计了光驱动的微型机器人(Zeng et al. 2018). 还可利用刚性塑料杆构建张拉整体机器人(Wang et al. 2019). 类似地, 剪纸结构也可用于设计滚动机器人(Cheng et al. 2020).
3.2.3 化学响应主动力学超材料
除了上述讨论的主动属性调制之外, 添加化学物质也可以在力学超材料的设计中增加多物理属性. 液体环境是化学响应材料所依赖的关键因素. 例如, 当pH、盐度、湿度或温度等外部化学物质驱动时, 水凝胶可以快速改变其形状. 水凝胶是一类柔软的亲水材料, 具有吸水和排水的潜能. 图案化的水凝胶在化学驱动下, 灵活多变, 具备可编程性(Palleau et al. 2013). Peng等(2016)开发了具有离子转移特性的印刷技术, 该技术可打印简单的水凝胶结构, 使其可以在更大的尺寸上修改为更复杂的结构形式. Hao等(2020)制造了基于剪纸结构的可编程复合水凝胶片. Wei等(2020)设计了具有负水合膨胀功能的拉胀超材料(参见图7(h)). Wang等(2014)提出了具有水诱导形状记忆效应的主动超材料. 研究表明, 物理和化学膨胀效应会影响水驱动的形状记忆过程. 使用具有不同膨胀特性的两层板设计的拉胀超材料(Liu et al. 2016), 当板吸收溶剂导致负膨胀时, 它们会发生面外弯曲.
金属阳离子的引入会产生不同类型的主动超材料. 这些阳离子增强了物理性能并可用于控制变形. Zhou等(2018)提出了一种具有多物理力学的折纸结构, 耦合了力学和化学双重效应. 其中, 折纸结构中的弯曲取决于阳离子的刚度错配. 二维周期性水凝胶片的变形, 可由溶液中阳离子的浓度控制(Wang et al. 2017). 此外, 由于主动超材料在不同pH环境下的可编程行为, pH驱动超材料也吸引了研究人员的兴趣. pH值的变化可以触发超材料从2D向3D发生构型变化(Bassik et al. 2010). 化学驱动超材料可应用于传感器、软体机器人和生物医学设备等领域(de Loos et al. 2005).
3.2.4 电响应主动力学超材料
电驱动超材料的工作原理是电场引发力学行为变化(Levine et al. 2021). 因此, 由于电、热和力耦合产生非常规的变形特性, 便会涉及到这些超材料的多物理性质. 它们大致可分为以下两种类型. 第一类, 力学变形是利用导体中电流产生的热能而驱动的, 因此, 本质上是热驱动超材料, 温度场是通过电场产生的. 另一类是由化学或物理反应引起的外部电驱动触发的, 这包括电活性聚合物、电化学刺激材料和离子聚合物-金属复合材料.
电热驱动材料方面的工作包括将电气化产生的热量用来控制变形. 由电热刺激驱动的折纸结构, 可以根据输入电压进行快速且可逆的折叠变化(Zhu et al. 2020). 超材料机器人可以通过将功能纤维与织物合并, 以不受限的方式改变其形状(Buckner et al. 2020). 在最近关于压电晶格超材料的工作中, 通过双级框架设计, 可利用电场的函数进行主动调制弹性特性(Singh et al. 2022b, Singh et al. 2021)(参见图7(d)(i) ~ 图7(d)(iii)). 在施加特定的电压和微观结构的几何参数下, 泊松比可显示正值或负值.
电化学效应可用来设计一类新的主动超材料(Levine et al. 2021, Li et al. 2021, Liu et al. 2021, Nick et al. 2020, Xia et al. 2019). 通过合金化和脱合金反应, 发现超材料的微观结构在电化学上排列一致(Xia et al. 2019). Liu等(2021)制备了纳米级电化学传动装置表面, 其中将传动装置组合成折纸机器人. 另一项研究, 则在超材料的单胞中使用了金属液体(Nick et al. 2020), 其中金属液体的阻力随着施加力而变化, 导致微流体通道变形. 电刺激的超材料可用于植物学领域, 将力学信号转化为电信号, 从而产生具有极端黏附性能的导电超材料(Li et al. 2021).
3.2.5 磁响应主动力学超材料
在不同的工程结构和工业应用中, 对变形材料的需求导致了新型力学超材料的发展, 例如由磁场驱动的磁力学泡沫或磁弹性晶格等超材料(Scarpa et al. 2004, Scarpa & Smith 2004). 该领域的一项研究表明, 在恒定磁场下, 拉胀磁力学聚氨酯泡沫具有不同的吸声系数(Scarpa et al. 2004, Scarpa & Smith 2004). 在外加磁场作用下, 这些磁响应主动超材料可以发生重构(Grima et al. 2013, Jackson et al. 2018, Montgomery et al. 2021, Ren et al. 2019, Singh et al. 2013). 通过在橡胶基体中嵌入可磁化夹杂物获得的3D打印拉胀结构, 在受到外部磁场作用时会收缩(Kim et al. 2018). 在另一项研究中, 将磁性物质引入弹性体中, 弹性体在大磁场下表现出耦合扭曲-屈曲行为(Tipton et al. 2012). 研究发现, 磁性夹杂物可以控制半刚性和刚性力学晶格的变形(Dudek et al. 2020, Dudek et al. 2018, Grima et al. 2013, Slesarenko, 2020).
磁力学超材料表现出更优异的抗冲击性能(Dudek K K et al. 2019)和波衰减特性(Schaeffer Marshall & Ruzzene 2015, Schaeffer M & Ruzzene 2015). 这可以推动他们在防护和阻尼装置与按需控制领域的应用. 由嵌入磁性颗粒的六边形单元制成的力学超材料, 同时具有负刚度和负泊松比(Hewage et al. 2016). 准二维超材料可应用于三维结构中(Dudek et al. 2020). 在纳米尺度上, 这些磁力学超材料可改变磁畴演化(Dudek M R et al. 2019, Raghunath & Flatau 2015). 最近的一项研究提出了一种创新的磁力学超材料, 可以小型化并具有形状编程和形状恢复功能(Galea et al. 2022). 在磁致伸缩晶格中无接触的框架下, 有效弹性模量的主动按需规划是可能的(Singh et al. 2022a). 根据外加磁场的不同, 相同的超材料可以表现为坚硬的金属或柔软的聚合物.
多年来, 硬磁软质主动材料因其具有可逆和快速的形状变化而引起了研究者的广泛关注(Kim et al. 2018, Lum et al. 2016). 在这些材料中, 硬磁颗粒嵌入在软弹性体基体中, 导致高剩余磁通和高矫顽力. 由于这些材料是柔性的, 可以承受很大的变形, 外加磁场和残余磁场的作用会导致非线性变形. 该领域的一项最新研究表明, 利用磁驱动超材料在梁水平上的多物理行为, 在力学载荷和磁场的共同作用下, 可获得晶格水平上的有效弹性特性(Sinha & Mukhopadhyay 2023)(参见图7(a) ~ 图7(b)). 这些主动力学超材料在生物医学设备(Kim Y et al. 2019)和软体机器人(Liu Y et al. 2017, Lum et al. 2016, Novelino et al. 2020)中具有广泛的潜在应用.
3.2.6 压力响应主动力学超材料
压力驱动的性能调制方法是应用最为广泛的驱动方法之一. 超材料的形状在压力的作用下发生变化, 压力可由膨胀或收缩引起. 最近在该领域的一项研究, 观察了柔性超材料在承受压力梯度时的力学行为变化. 在这项工作中, 气动驱动的概念与超材料的设计模式相结合, 产生了调节效果(Narang et al. 2018). 据报道, 一种压敏弯曲软超材料具有拉胀性能(Pan et al. 2020). 有学者发现, 具有周期性孔排列的超材料, 其中孔的形状可以气动定制, 可导致力学行为的变化(Chen & Jin 2018). 通过引入像气球一样膨胀的空腔, 可实现具有可调节负刚度的超材料(Tan et al. 2020, Tan et al. 2019). 由于具有折叠和可展开的优点, 气动驱动方法适用于折纸和剪纸结构(Kim W et al. 2019, Li S et al. 2019). 通过双级框架设计, 可在充气网格中实现极端比刚度. 这里利用充气式梁的基本力学特性, 得到了晶胞级有效弹性特性. 这些网格可以应用于不同的结构中, 满足可部署性、存储性和可移植性的要求(Sinha & Mukhopadhyay 2023b)(参见图7(b) ~ 图7(c)), 并对弹性模量进行主动控制. 其他研究利用流体的黏度来设计液压驱动的主动超材料, 从而可以在所有气动驱动的超材料设计中使用(Chen S et al. 2020). 其中一项工作提出了一种流体折纸, 其刚度可以根据流体的体积来调整(Li & Wang 2015). 当压力驱动的超材料与其他物理参数相结合时, 可获得前所未有的新特性, 如水凝胶传动装置中的自主排汗(Mishra et al. 2020).
4. 力学超材料的物理实现
由于算力的提升, 设计工具的精进, 催生了力学超材料更为复杂的微观结构(Gu et al. 2018b, Gu et al. 2017, Luo et al. 2020, Mao et al. 2020, Wu et al. 2020). 结构的几何裁剪是在微观甚至纳米尺度上完成的, 传统的减材制造方法已不能满足需求. 此外, 微观结构可以由多种材料组成(Mirzaali et al. 2018, Mukhopadhyay et al. 2020c). 因此, 研究人员开始使用3D打印或增材制造来实现复杂的微观结构 (见图8). 增材制造的进步使亚微米参数的制造成为可能, 这在以前是不可能的(Greer & Deshpande 2019, Mao et al. 2017, Schaedler & Carter 2016). 此外, 多材料增材制造(Kokkinis et al. 2018, Kokkinis et al. 2015, Kuang et al. 2019b, Roach et al. 2019, Xu et al. 2019)本质上提供了比传统制造技术更大的设计空间. 3D打印的最新进展也使刺激响应材料的制造成为可能, 从而产生具有可转换功能的超材料, 包括几何形状或物理属性. 这通常称为用于制造主动力学超材料的4D打印(Ge et al. 2014, Ge et al. 2013, Kuang et al. 2019a)(参见图8(f)). 在下面的小节中, 我们将简要介绍用于制造力学超材料的3D打印技术.
4.1 材料挤出
这种制造方法包括熔融沉积建模(FDM)和直接墨水书写(DIW). 由于其方法简单, 是制造力学超材料最流行的方法之一. FDM广泛用于打印工程热塑性塑料(Ligon et al. 2017, Ngo et al. 2018, Surjadi et al. 2019)(参见图8(a)(i) ~ 图8(a)(iii))和颗粒嵌入复合材料领域, 这些材料对精度要求不高. 在这种方法中, 使用固体聚合物长丝在构建板上分层绘制二维切片. 每一层逐层沉积, 直到获得完整的3D部件. DIW使用黏弹性或黏塑性油墨, 在喷嘴上施加压力后沉积, 以绘制二维切片(Grosskopf et al. 2018, Robertson et al. 2018)(参见图8(a)(iv) ~ 图8(a)(vi)). DIW广泛用于打印各种多孔微结构, 包括梁或支柱. 材料挤出过程是一种缓慢的方法, 每层沉积一次形成一层, 因为从一维线开始打印. 图8(a)(ii)为采用该方法打印的三维阵列桁架结构样品(Kaur et al. 2017). 这些结构的支柱厚度保持在最薄的可打印尺寸, 以获得最小的特征. 同样, 通过化学镀方法在纯聚合物晶格上连续沉积一层薄薄的镍磷合金, 可得到涂镍聚合物介晶格复合材料(见图8(a)(iii)) (Song et al. 2018).
4.2 喷墨印刷
喷墨打印(IJP)用于多材料制造(Ding et al. 2017). 在这种方法中, 使用尺寸从20到40 μm不等的墨滴, 通过压电喷嘴沉积在构建板上(参见图8(b)(i)). 所需的结构是通过打印头在快速通道中平移整个构建板来获得的. IJP使用光固化油墨. IJP的一种高级形式是PolyJet技术, 用于打印数字材料(Tee et al. 2020)(参见图8(b)(ii)). 该技术使用刚性和软树脂油墨, 它们以固定的比例沉积, 有助于确定最终成型材料的有效性能(参见图8(b)(iii) ~ 图8(b)(v)). IJP不适合打印多孔结构或晶格, 因为墨滴会单独沉积, 导致需要额外的支撑材料. 还要求油墨具有低黏度, 从而限制了印刷中嵌入颗粒新材料. 图8(b)(iii)展示了使用细胞结构(使用PolyJet打印制造)来预测屈曲模式的实验(Janbaz et al. 2019), 其中的几何缺陷是由用于预先处理胞元结构的屈曲模态对应的变形累积产生的. PolyJet打印也可用于创建物理逻辑门(Waheed et al. 2020)和具有韧性可调的复合材料(Lei et al. 2018)(参见图8(b)(iv) ~ 图8(b)(v)).
4.3 光聚合固化
光固化技术是使用不同波长的光来固化光敏性聚合物树脂的一种方法. 该方法包括双光子聚合(TPP)、数字光处理(DLP)和立体光刻(SLA). SLA是通过紫外线激光束扫描固化树脂的一种增材制造方法. 它可以大面积打印, 但速度可能很慢. DLP通过简单的低成本设置, 提供高打印速度(Chen & Zheng 2018)(参见图8(c)(i) ~ 图8(c)(iii)), 一次固化一层. 使用微镜装置的投影仪将2D光的图案照射到树脂桶中. TPP方法使用激光束在树脂中绘制形状(Serbin et al. 2004)(参见图8(c)(iv) ~ 图8(c)(vi)). 在这种方法中, 固化过程是使用一个非常高强度的激光焦点, 促使两个光子的吸收. 这允许激光束穿透树脂桶, 留下上部区域未固化. 图8(c)(ii)为采用μDLP方法打印的具有不同材料的3D打印微晶格结构(Chen & Zheng 2018). 图8(c)(v)显示了用TPP打印的单元格高度为18 μm、光束直径为0.5 μm的微晶格(Vangelatos et al. 2019).
4.4 粉床熔融
粉末床熔融(PBF)是一种不同于其他方法的3D打印方法, 包含电子束熔化(EBM)(参见图8(d)(iii) ~ 图8(d)(iv))(Parthasarathy et al. 2010, Rafi et al. 2013), 选择性激光熔化(SLM)(Kruth et al. 2004, Thijs et al. 2010)(参见图8(d)(i) ~ 图8(d)(ii))和多射流熔融(MJF)(O’Connor et al. 2018)等技术. 在这些方法中, 光源或高功率激光加热打印床, 打印床上沉积有一层薄粉层(Yang et al. 2015). 粉末层根据所需的二维图案形状进行加热, 引发材料损耗. PBF有益于制造复杂的结构, 如空心晶格、锥形壁晶格等, 这些结构通常难以注塑或机械加工成型. 图8(d)(ii)显示了使用SLM制造的五模超材料. 晶格在每个方向上有5个单元格, 标称支撑尺寸为3.464 mm (Hedayati et al. 2017). 图8(d)(iv)为用EBM制备的晶格. 这些格子被设计成具有4 × 4 × 4的单胞重复次数(Yang et al. 2015).
5. 力学超材料的发展趋势和未来路线图
尽管力学超材料作为一个公认的科学领域的历史不超过十年或二十年, 而且它还在不断发展, 但这一时期的激烈研究并没有为未来的研究提供明确的趋势和方向. 在本节中, 我们将讨论这些趋势, 以及在不久的将来需要超材料界进一步关注的领域.
5.1 复杂微结构的4D打印
力学超材料领域的广泛研究使得突破传统材料性能的极限成为可能. 3D打印技术使得通过物理方式实现复杂的微结构设计成为可能, 这在以前使用传统制造方法是不可能实现的. 力学超材料的新兴趋势之一是物理性质和形状按需主动调制. 这是通过涉及主动材料的4D打印实现的. 4D打印仍处于发展的初级阶段, 需要更多的关注, 例如如何提高涉及多组分的不同主动材料的精度和功能. 生物4D打印是一个即将到来的研究方向, 利用生物的存在来主动控制物理性质, 包括结构的损伤修复和再生.
5.2 实时可重构和功能编程
被动力学超材料在制造后不允许进行性能调制. 然而, 主动力学超材料允许在制造后利用外部刺激(如电场或磁场)来控制静态和动态的性能参数. 根据实际情况, 可实时按需变形, 实现有效特性的可编程调制. 以折纸和剪纸为基础的超材料, 可允许大规模的形状变化以及有效本构行为的接触诱导编程. 实时可重构和功能编程的概念是相当新的, 在这个方向上有必要进行大量的研究, 以探索主动变形和单胞级力学的交互空间, 以及特定应用产品的开发. 在此背景下, 超材料的多物理行为需要进一步关注, 以满足现代结构系统的多功能需求. 例如, 能量收集超材料应该能够有效承载力学负荷并储存能量以备将来使用, 从而实现最优的材料利用率.
5.3 纳米级超材料
大多数关于超材料的研究主要集中在微观尺度上, 以获得宏观尺度上有效的物理性质. 然而, 随着纳米制造的最新进展, 在纳米尺度上复制这些概念存在着巨大的可能性. 科学界最近开始探索纳米尺度的结构, 这一领域在计算和实验创新方面都有很长的路要走.
5.4 可扩展性
在工业规模上采用超材料的一个紧迫挑战是可扩展性和大规模生产的问题, 并有充分的质量控制. 大多数关于超材料的实验研究, 依赖于相对小尺寸的物理样本. 从理论上讲, 如果尺寸增加, 周期性微结构的有效性能不会发生任何变化, 但利用超材料制造大型结构(即大构建体积)仍然是一个挑战. 机器人增材制造和无人机辅助增材制造, 可能是解决此类问题的前瞻性技术路线(见图9(c)). 此外, 通过力学键连接超材料的多个片段(或单胞)可能是构建大型结构的另一种有希望的良策(参见图9(a) ~ 图9(b)). 由于增材制造已经成为最适合制造复杂超材料微结构的技术之一, 除了实现工业级的大规模生产外, 还有其他挑战有待解决, 如生产速度慢、材料属性不一致、增材制造后处理的自动化以及初始投资的要求等等.
5.5 人工智能和机器学习
随着力学超材料设计的日趋复杂, 对其响应进行预编程的需求日益突出. 近年来, 借助人工智能(AI)和机器学习(ML)设计人工微观结构, 吸引了科学界的关注. 通过人工智能和机器学习, 我们可以在超材料的设计中进行几何优化, 实现多功能. 在单元格拓扑优化中, 可以采用逆设计方法, 在给定约束条件下实现多个目标, 这是传统的基于正向直观框架的单元格设计方法所无法实现的. 近年来, 力学超材料领域的研究已经从分析合理的微观结构发展到开发有效的计算方法(Ma et al. 2001, Nabian & Meidani 2018). 为了实现这一目标, 人工智能已应用于设计最佳材料和微结构当中(Dima et al. 2016, Jain et al. 2013, Kirklin et al. 2015, Neelakantan et al. 2014, Sundararaghavan & Zabaras 2005, Wang et al. 2013). 人工智能和机器学习为微观结构的合理设计提供了高精度和便利性, 从而激发了面向应用的多功能力学超材料(Goldsmith et al. 2018, Helma et al. 2004, Liu R et al. 2017, Ward et al. 2016).
多年来, 力学超材料领域的研究涉及使用解析和实验研究, 或使用复杂而微妙的有限元模拟研究. 有限元方法计算量大, 耗时长, 而实验研究涉及繁重的操作. 考虑到单胞结构及其尺寸的巨大可能性, 通过单独建模每个微观构型, 或通过制造它们进行实验测试来进行有限元模拟实际上是不可能的. 对于结构复杂性较小的晶格, 解析性研究通常是首选的. 因此, 研究人员现在将重点放在机器学习上, 将其作为替代模型, 使各种复杂的超材料构型更容易处理. 这些模型可以通过建立有效的计算映射将超材料的输入(如微观结构设计)和输出参数(如有效力学性能)联系起来(Isanaka et al. 2022). 近年来机器学习的进步引发了不同相关领域的进步, 不仅包括超材料和高级复合材料的设计(Bessa et al. 2019, Gu et al. 2018b, Gu et al. 2018a, Guo et al. 2021, Ma et al. 2018, Sharma et al. 2022, Wilt et al. 2020), 而且为优化制造方法铺平了道路(Pahlavani et al. 2022, Wang C et al. 2020, Wang et al. 2021). 基于图像的机器学习方法(Gupta et al. 2023), 可以通过捕获单胞结构进一步增强对有效力学性能预测的泛化能力.
在超材料的多尺度设计背景下, 如图2所示, 材料属性在两个不同的尺度上定义. 机器学习在两个层面上都是有用的: (1)通过开发原子间电位来定义较低尺度上的固有材料性质(Mishin 2021, Mortazavi et al. 2023, Mueller et al. 2020); (2)如上文所述, 在更大尺度上进行单胞水平设计.
神经形态是力学超材料研究中一个有趣而新颖的领域(Xia et al. 2022). 这意味着可以在结构和功能上复制这种超材料, 以发挥生物神经网络的作用. 力学超材料的两个特性, 即分层连接性和网络中每个节点对的加权耦合, 使其对神经形态有用. 随着人工智能的发展, 这一领域的探索将开辟广阔的研究空间.
5.6 多物理折纸/剪纸
基于折纸和剪纸的超材料由于其可编程的特性, 引起了极大的关注. 研究界也开始意识到, 在这些超材料中加入主动成分以增强可编程性的潜在优势. 这些基于多物理折纸/剪纸的超材料将激发基于外部刺激的主动特性调制, 包括非接触的驱动方式.
5.7 具有生命物质的微结构
以生命物质为构成元素的超材料是一个即将到来的研究方向, 其中生物体的存在将用于主动控制物理特性和结构的程序化生长. 值得注意的是, 编程增长不仅可以解决可扩展性问题, 而且还有助于实现不同的曲率和预定义的形状. 此外, 这种超材料具有自愈和损伤修复的能力.
5.8 柔性和共形超材料
力学超材料中发展最快的领域之一是柔性超材料, 其预期应用在包括软体机器人和生物医学设备在内的一系列工程应用中. 在这种体系中, 非线性和大变形方面变得突出, 科研界越来越重视这一点. 此外, 最近的研究表明, 通过引入多物理力学, 可以将柔性力学超材料主动转换为硬质材料. 这种多功能的按需属性调制值得更多的关注. 柔性超材料的形状顺应性以及主动适应预定义形状将开辟大量创新应用.
5.9 制作缺陷的影响
增材制造已成为实现复杂超材料微结构的最重要方式. 然而, 增材制造通常存在一系列缺陷, 例如制造的几何形状中存在非预期的预应力、空隙和不规则性. 这种效应可以显著影响超材料的有效特性. 此外, 材料的固有属性取决于增材制造的类型和制造过程中不同的其他校准参数. 需要在这些方向上进行深入研究, 以量化制造缺陷对增材制造其他不确定性的影响.
5.10 使用寿命影响: 环境和工作条件
除了制造不确定性之外, 力学超材料的使用寿命条件也需要引起高度关注. 根据数字孪生的概念, 超材料的设计应考虑周围环境、材料退化和随时间推移的损伤累积的综合影响. 此外, 还有一些关键的长期力学性能, 如疲劳、黏弹性和蠕变的影响, 以及增材制造方法对失效模式的影响, 这些方面尚未得到足够的关注. 为了提高在工业结构中利用这些复杂微观结构的可靠性, 需要在这些方向上进一步研究.
6. 结论
力学超材料领域的广泛研究使得设计具有非常规行为的材料成为可能, 这些材料具有相互冲突且通常不相关的多物理目标, 从而扩大了材料设计空间, 突破了物理性能极限. 在本文中, 我们回顾了力学超材料的最新进展, 重点强调了主动和多物理行为 (涉及外部电场或磁场, 以及温度、光或化学反应等刺激) 以及双级结构的力学行为, 以扩大主动编程按需响应的范围. 我们首先简述了基于功能和微观构型分类的超材料, 然后对被动和主动力学超材料进行了全面综述. 值得注意的是, 随着力学超材料设计中多功能性和按需属性调制的出现, 不同超材料类别之间的划分变得相互关联, 微结构设计需要越来越多的交互空间. 接着, 对力学超材料领域的发展趋势和未来路线图进行了全面分析, 包括实时可重构和功能编程、4D 打印、纳米级超材料、可扩展性、人工智能和机器学习、神经形态、多物理折纸/剪纸等概念、生命物质、柔性和共性超材料、制造和使用寿命影响. 本文对涉及力学超材料各个新兴方面的文献和应用观点进行了全面的回顾总结, 为研究人员和工程师探索新趋势、新模式和多物理设计空间提供了丰富的可能性, 从而开发具有创新性和革命性的新型超材料.
竞争利益声明 作者声明, 他们没有已知的可能会影响本人报告的竞争性财务利益或个人关系
数据可用性 数据将应要求提供.
致谢 感谢印度教育部通过博士奖学金提供的财政支持. 感谢在研究工作期间从南安普敦大学获得的启动资金.译者感谢北京市海淀联合创新基金 L212017资助.
-
图 1 材料和结构在超材料中相互作用. (a)在材料尺度上利用力学设计和人工微结构来提高性能(Schaedler & Carter 2016). 随着技术的进步, 功能结构的定义变得更加复杂, 并从宏观走向更精细的微观和纳米尺度. (b)跨尺度点阵结构的适用性和工程应用(Mukhopadhyay & Adhikari 2017a). (c)泊松比可编程的六边形晶格微结构: (i)正泊松比; (ii)负泊松比; (iii) ~ (v)零泊松比(Mukhopadhyay & Adhikari 2017b). (d)超平面弯曲的高斯曲率编程结构: (i)负; (ii)正(Mirzaali et al. 2021). (e) Sarrus模块化折纸设计: (i)描述Sarrus机制的立方单位; (ii) Sarrus连杆的装配; (iii) ~ (v)模块变换顺序; (vi) ~ (viii) Sarrus超材料(Yang & You 2020)的变形构型. (f)基于折纸/剪纸的模块材料: (i)一个典型的截断八面体; (ii) ~ (iii)由截断的八面体衍生的对称和非对称单胞; (iv)由对称单元格堆叠得到二维元表面; (v)承受压缩载荷时的超板材结构; (vi)受拉伸载荷(Sinha & Mukhopadhyay 2022c)时的板材超结构. (g)基于水弹折纸的管状超材料: (i)基于水弹折纸的折痕图; (ii)水弹管的3D视图; (iii)管状元结构的宏观图示; (iv)微观结构和远场驱动相关的形状变形; (v)基于微观结构的本构关系编程(Mukhopadhyay et al. 2020a)
图 3 多功能超材料总览 (a)电磁超材料: (i)由铜制分裂环谐振器和导线构成的2维周期阵列组成一种“左手材料”(LHM), 呈现出负折射(Shelby et al. 2001); (ii)圆形、方形、单环和多环结构的分裂环谐振器, 分裂环谐振器(SRR)是一种高导电性结构, 其电感由两个环之间的电容平衡; (iii)瑞士卷结构的透视图和俯视图, 瑞士卷结构内的电流是由结构的自身电容引起的, 能够形成完整交流电路; (iv)圆锥形瑞士卷的侧视图和俯视图, 圆锥形瑞士卷结构有助于电磁波在相对较大的距离上传输, 同时减少了阻尼; (v)用于制作手性瑞士卷结构的导电片展开结构和手性瑞士卷结构的俯视图, 导电片的每一层都填充有介质材料(Grimberg 2013). (b)光学超材料: (i)常规透镜聚焦光线; (ii)负折射率超材料; (iii)给出负电响应的等离子体纳米棒; (iv)提供负磁共振的分裂纳米环; (v)与双分裂纳米环配对的纳米棒, 产生负的磁和电响应; (vi)耦合纳米棒在特定情况下也能呈现负的磁和电响应; (vii)由双开口环组成的3D光学超材料单胞(Gardner et al. 2011). (c)声学超材料: (i)带有侧孔图案的管状结构, 显示出负有效模量, 单胞结构如侧图显示; (ii)测量相位和传输速度的实验装置(Lee et al. 2009b). (d)力学超材料: 显示出拉胀性和可逆变形的微观点阵(Schaedler & Carter 2016)
图 4 力学超材料的实际应用 (a)隐身斗篷(Bückmann et al. 2014). (b)电子皮肤(Li K et al. 2019). (c)软体机器人(Cheng et al. 2020). (d)仿生夹具(Wang X Q et al. 2020). (e)通过超材料进行力学计算(Zhang et al. 2023). (f)抗冲击结构(Evans & Alderson 2000). (g)血管支架(Jia et al. 2018). (h)跑鞋(Gleeson 2020). (i)柔性电池(Bao et al. 2020). (j)桁架芯翼型(Spadoni & Ruzzene 2007). (k)拉胀安全带(Balan P et al. 2023). (l)海洋防护结构应用中的组装蜂窝结构(Lang et al. 2023). (m)拉胀绷带(Balan P et al. 2023). 值得注意的是, 我们在这里只提到了几个有代表性的应用, 实际上这个列表是无穷无尽的
图 5 超材料的微结构构型 (a)不同形状的晶格: (i)规则的三角形网格; (ii)正六边形晶格; (iii)被称为Kagome的半正三角形-六边形晶格(Ongaro 2018); (iv)具有不同类型组成元素的晶格超材料, 例如基于板、基于支柱/梁和基于TPMS的构件(Zhong et al. 2023). (b)手性和反手性蜂窝(Ongaro 2018): (i)三手性蜂窝(Alderson et al. 2010); (ii)四手性蜂窝; (iii)六手性蜂窝; (iv)反三手性蜂窝(Alderson et al. 2010); (v)反四手性蜂巢. (c)基本变形模式: (i)弯曲占优势的晶格晶胞; (ii)受拉伸支配的晶格单胞(Zheng et al. 2014). (d)基于折纸的力学超材料: (i) Miura-ori层的运动行为; (ii) Miura-ori折纸结构的单胞几何形状(Schenk & Guest 2013). (e)混合折纸超材料: (i)标准Miura-ori图案化的晶胞, 可表现出面内拉胀性; (ii)混合折纸单元的3D视图, 由常规凹六边形蜂窝单元和常规Miura-ori图案组合而成; (iii)通过周期性重复单胞获得的超材料微结构(Wang H et al. 2020). (f)基于折纸和剪纸不同类别的超材料: (i)刚性折纸; (ii)刚性混合折纸/剪纸; (iii)刚性剪纸; (iv)可变形折纸; (v)可变形的混合折纸/剪纸; (vi)可变形的剪纸; (vii)基于折纸和剪纸的超材料的子类(Zhai et al. 2021)
图 6 力学超材料的被动特性调制 (a)基于单胞自下而上的方法对蜂窝结构分析: (i)规则的2D蜂窝状晶格; (ii)堆叠时形成整个晶格的蜂窝的单胞; (iii)蜂窝的单元壁, 将其视为具有自由度的梁单元(P Sinha & Mukhopadhyay 2022). (b)规则的3D蜂窝晶格(Sinha et al. 2023a). (c) 2D六边形蜂窝结构的力学特性: (i)未变形的蜂窝; (ii)沿X1方向承受面内载荷的单胞; (iii)沿X2方向承受面内载荷的单胞; (iv)承受面内剪切载荷的单胞(Andrews et al. 1999). 对于其他2D和3D晶格, 需要适当分析单胞的力学特性. (d)不同应力条件下晶格中的抗弯曲效应: (i)正六边形晶格的单胞; (ii)在经受沿X1方向的拉伸载荷的压应力时, 具有弯曲胞壁的单胞; (iii)当受到剪切应力时获得的具有弯曲细胞壁的单胞; (iv)针对X1方向上的拉伸正应力, 蜂窝网格单元壁中具有的反曲率; (v)蜂窝晶格单元壁在抗逆时针剪应力具有的反曲率(Ghuku & Mukhopadhyay 2023). (e) 2D Voronoi蜂窝的单元不规则性: (i)未变形构型中的随机Voronoi蜂窝; (ii)具有周期性边界条件的变形Voronoi蜂窝结构(Zhu et al. 2001). (f)波在力学超材料中的传播: (i)共振球形原子的横截面; (ii)声波晶体中具有单胞的声学超材料(Lu et al. 2009). (g)具有区域不连续性的典型非不变蜂窝晶格(Sinha et al. 2023b). (h)基于纳米异质结构的纳米级超材料概念: (i)典型纳米结构的侧视图和俯视图, 其中单一类型的原子在单一平面中形成整个结构; (ii)典型纳米结构的侧视图和俯视图, 其中不同类型的原子在单个平面中形成整个结构; (iii)典型纳米结构的侧视图和俯视图, 其中原子在多个平面中形成整个结构; (iv)典型纳米结构的侧视图和俯视图, 其中不同类型的原子在多个平面中形成整个结构; (v)由不同2D材料组成的多层异质结构(Mukhopadhyay et al. 2020b)
图 7 力学超材料的主动特性调制 (a) ~ (d)双能级主动晶格超材料: (b)中所示的晶格(拉胀型和非拉胀型)由(a)、(c)和(d)中所示的不同主动梁组件制成; 基于(a)中所示的硬磁软(HMS)梁形成磁驱动主动力学超材料, 图中显示了外磁驱动下未变形和变形配置的HMS梁(Sinha & Mukhopadhyay 2023); 压力驱动的主动力学超材料是基于(c)所示的可充气梁形成的, 在这种可膨胀的晶格中可以进行卷绕和密实堆叠(P Sinha et al. 2023); 电驱动主动力学超材料基于压电复合梁形成, 如(d)所示, 其中通过单晶形和双晶形构型可以实现纯弯曲、纯拉伸和弯曲/拉伸组合模式(Singh et al. 2022b). (e)光驱动主动力学超材料: (i)光聚合后的聚合物片; (ii)空间的、 不同固化板的自由弯曲; (iii)用后固化法在均匀光线下对弯曲结构进行定形; (iv)后固化后的硬样品; (v)花结构的不同开放程度; (vi)聚合物板材曲率的连续变化(Zhao et al. 2017). (f) 3D光学力学超材料: (i)使用外部准静态电场
$ \overrightarrow{E} $ 来定向光敏树脂的液晶方向, 然后通过双光子聚合(TPP)打印的体素进行局部固定; (ii)结构的另一部分具有不同的方向排列, 导致电场的排列发生变化; (iii)结构的聚合和未聚合区域; (iv)包含定向3D液晶导向场的光学透明聚合结构; (v)样品浸泡在染料溶液中, 染料溶液扩散到样品中, 并作为吸收剂与刺激光耦合; (vi)最终超材料(Münchinger et al. 2022). (g)受光刺激的力学超材料: 基于晶格的结构, 当外部LED关闭和打开时, 分别显示正的和负的(i)泊松比和(ii)扭转应变(Münchinger et al. 2022). (h)化学驱动的主动力学超材料: (i) 2D超材料在水合和脱水时的变形模式; (ii) 3D超材料的负水化膨胀变形(Wei et al. 2020). (i)热驱动主动力学超材料: (i)热致动超弹性超材料的变形; (ii)弹性薄层压板向体积材料的转变; (iii)超材料的大变形, 卸载后可恢复其形状(Wu et al. 2021)图 8 力学超材料的物理实现 (a)材料挤压工艺: (i)熔融沉积建模 (FDM) 的设置和工作方法演示 (Surjadi et al. 2019); (ii)基于FDM的聚合物八重桁架晶格(Kaur et al. 2017); (iii)通过化学镀和FDM制造的具有Ni外壳和聚合物芯的复合晶格(Song et al. 2018); (iv)直接墨水书写(DIW)的基本工作方法说明 (Surjadi et al. 2019); (v)一种基于3D微点阵的DIW (Rozvany 2009); (vi)使用DIW (Compton & Lewis 2014)制造的纤维填充环氧蜂窝复合材料. (b)喷墨打印: (i)喷墨打印设置示意图(IJP) (Ding et al. 2017); (ii)使用Polyjet打印尺寸和形状可调的颗粒复合材料(Tee et al. 2020); (iii)由于使用Polyjet在周期性晶格中插入刚性缺陷, 导致周期性晶格屈曲(Janbaz et al. 2019); (iv)使用PolyJet印刷制造的物理与门, 允许逻辑可调属性(Waheed et al. 2020); (v)通过改变数字材料的比例来控制2D复合材料 (PolyJet打印) 的断裂特性(Lei et al. 2018). (c)光固化技术: (i)数字光处理(DLP)技术图示(Chen & Zheng 2018); (ii)使用多材料μDLP方法(Chen & Zheng 2018)创建具有可定制泊松比的超材料; (iii)由于光固化技术的灵活性, 晶格的愈合得以实现(Yu et al. 2020); (iv)双光子聚合 (TPP) 技术说明(Serbin et al. 2004); (v) TPP可以产生具有可调屈曲特性的微晶格(Vangelatos et al. 2019); (vi) TPP工艺可以导致更大的晶格间距, 因为它允许结构收缩, 从而使结构显示出光子性质(Liu Y et al. 2019). (d)粉末床熔融: (i)选择性激光熔化(SLM)装置演示(Surjadi et al. 2019); (ii)使用SLM印刷的五模超材料(Hedayati et al. 2017); (iii)电子束熔化(EBM)装置演示(Surjadi et al. 2019); (iv)使用EBM制造的晶格(Yang et al. 2015). (e)六边形晶格结构的运动动力学, 用于将其从平面配置折叠. 晶格结构的设计基于重复晶胞(Janbaz et al. 2017). (f)利用4D打印制造主动超材料, 并实现主动的时间依赖性和可编程响应(Boley et al. 2019, Kim et al. 2022, Zeng et al. 2022). 这本质上是通过耦合3D打印和主动材料来制造时间依赖性和外部刺激敏感的超材料来实现的
图 9 力学超材料的可扩展性和大型建造体积增材制造的途径. (a)通过组装过程制造超材料(Lang et al. 2023). 这里, 每个超材料晶胞(或一组几个晶胞)可以额外制造并随后组装以实现大的构建体积. (b)离散组装的力学超材料. 从左到右显示了四种不同类型的超材料, 即刚性、柔性、拉胀性和手性. (i)端面轮廓; (ii)单个体素的前视图; (iii) 2 × 2 × 2立方体的前视图; (iv)单个体素的斜视图; (v) 2 × 2 × 2斜视图(Jenett et al. 2020). (c)用于实现大构建体积和规模的常规、机器人和无人机辅助(空中)增材制造(Zhang et al. 2022)
-
[1] Abd El-Sayed F K, Jones R, Burgess I W. 1979. A theoretical approach to the deformation of honeycomb based composite materials. Composites, 10: 209-214. doi: 10.1016/0010-4361(79)90021-1 [2] Adhikari S, Mukhopadhyay T, Liu X. 2021. Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach. Mechanics of Materials, 157: 103796. doi: 10.1016/j.mechmat.2021.103796 [3] Adhikari S, Mukhopadhyay T, Shaw A, Lavery N P. 2020. Apparent negative values of Young’s moduli of lattice materials under dynamic conditions. International Journal of Engineering Science, 150: 103231. doi: 10.1016/j.ijengsci.2020.103231 [4] Aghighi F, Morris J, Amirkhizi A V. 2019. Low-frequency micro-structured mechanical metamaterials. Mechanics of Materials, 130: 65-75. doi: 10.1016/j.mechmat.2018.12.008 [5] Alderson A, Alderson K L, Chirima G, Ravirala N, Zied K M. 2010. The in-plane linear elastic constants and out-of-plane bending of 3-coordinated ligament and cylinder-ligament honeycombs. Composites Science and Technology, Special issue on Chiral Smart Honeycombs, 70: 1034-1041. [6] Al-Mulla T, Buehler M J. 2015. Folding creases through bending. Nature Mater., 14: 366-368. doi: 10.1038/nmat4258 [7] Alturki M, Burgueño R. 2019. Multistable cosine-curved dome system for elastic energy dissipation. Journal of Applied Mechanics, 86 : 091002 [8] Alù A, Engheta N. 2008. Dielectric sensing in $ \mathrm{\epsilon } $ -near-zero narrow waveguide channels. Phys. Rev. B, 78: 1098-1121.[9] Ambati M, Fang N, Sun C, Zhang X. 2007. Surface resonant states and superlensing in acoustic metamaterials. Phys. Rev. B, 75: 195447. doi: 10.1103/PhysRevB.75.195447 [10] Andrews E W, Gibson L J, Ashby M F. 1999. The creep of cellular solids. Acta Materialia, 47: 2853-2863. doi: 10.1016/S1359-6454(99)00150-0 [11] Ashby M F. 2005. The properties of foams and lattices. Philosophical Transactions of the Royal Society A: Mathematical. Physical and Engineering Sciences, 364: 15-30. [12] Bacquet C L, Al Ba’ba’a H, Frazier M J, Nouh M, Hussein M I. 2018. Metadamping: Dissipation emergence in elastic metamaterials. Advances in Applied Mechanics, 51: 115-164. [13] Bakhvalov N, Panasenko G. 1989. Homogenisation: Averaging processes in periodic media: Mathematical problems in the mechanics of composite materials. Netherlands, 36 . [14] Balan P M, Mertens A J, Bahubalendruni M V A R. 2023. Auxetic mechanical metamaterials and their futuristic developments: A state-of-art review. Materials Today Communications, 34: 105285. doi: 10.1016/j.mtcomm.2022.105285 [15] Banerjee A, Das R, Calius E P. 2017. Frequency graded 1d metamaterials: A study on the attenuation bands. Journal of Applied Physics, 122: 075101. doi: 10.1063/1.4998446 [16] Bao Y, Hong G, Chen Y, Chen J, Chen H, Song W-L, Fang D. 2020. Customized kirigami electrodes for flexible and deformable lithium-ion batteries. ACS Appl. Mater. Interfaces, 12: 780-788. doi: 10.1021/acsami.9b18232 [17] Bassik N, Abebe B T, Laflin K E, Gracias D H. 2010. Photolithographically patterned smart hydrogel based bilayer actuators. Polymer, 51: 6093-6098. doi: 10.1016/j.polymer.2010.10.035 [18] Bauer J, Schroer A, Schwaiger R, Kraft O. 2016. Approaching theoretical strength in glassy carbon nanolattices. Nature Mater., 15: 438-443. doi: 10.1038/nmat4561 [19] Benedetti M, du Plessis A, Ritchie R O, Dallago M, Razavi N, Berto F. 2021. Architected cellular materials: A review on their mechanical properties towards fatigue-tolerant design and fabrication. Materials Science and Engineering: R: Reports, 144: 100606. doi: 10.1016/j.mser.2021.100606 [20] Bertoldi K, Vitelli V, Christensen J, van Hecke M. 2017. Flexible mechanical metamaterials. Nat. Rev. Mater., 2: 1-11. [21] Bessa M A, Glowacki P, Houlder M. 2019. Bayesian machine learning in metamaterial design: Fragile becomes supercompressible. Advanced Materials, 31: 1904845. doi: 10.1002/adma.201904845 [22] Bigoni D, Guenneau S, Movchan A B, Brun M. 2013. Elastic metamaterials with inertial locally resonant structures: Application to lensing and localization. Phys. Rev. B, 87: 174303. doi: 10.1103/PhysRevB.87.174303 [23] Blees M K, Barnard A W, Rose P A, Roberts S P, McGill K L, Huang P Y, Ruyack A R, Kevek J W, Kobrin B, Muller D A, McEuen P L. 2015. Graphene kirigami. Nature, 524: 204-207. doi: 10.1038/nature14588 [24] Boatti E, Vasios N, Bertoldi K. 2017. Origami metamaterials for tunable thermal expansion. Advanced Materials, 29: 1700360. doi: 10.1002/adma.201700360 [25] Boley J W, van Rees W M, Lissandrello C, Horenstein M N, Truby R L, Kotikian A, Lewis J A, Mahadevan L. 2019. Shape-shifting structured lattices via multimaterial 4D printing. Proceedings of the National Academy of Sciences, 116: 20856-20862. doi: 10.1073/pnas.1908806116 [26] Brillouin L N. 1953. Wave propagation in periodic structures: Electric filters and crystal lattices. New York: Dover Publications. [27] Brunet T, Merlin A, Mascaro B, Zimny K, Leng J, Poncelet O, Aristégui C, Mondain-Monval O. 2015. Soft 3D acoustic metamaterial with negative index. Nature Mater., 14: 384-388. doi: 10.1038/nmat4164 [28] Bückmann T, Thiel M, Kadic M, Schittny R, Wegener M. 2014. An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun., 5: 4130. doi: 10.1038/ncomms5130 [29] Buckner T L, Bilodeau R A, Kim S Y, Kramer-Bottiglio R. 2020. Roboticizing fabric by integrating functional fibers. Proceedings of the National Academy of Sciences, 117: 25360-25369. doi: 10.1073/pnas.2006211117 [30] Cai R, Jin Y, Rabczuk T, Zhuang X, Djafari-Rouhani B. 2021. Propagation and attenuation of rayleigh and pseudo surface waves in viscoelastic metamaterials. Journal of Applied Physics, 129: 124903. doi: 10.1063/5.0042577 [31] Cai W, Chettiar V, Kildishev A, V Shalaev. 2007. Optical cloaking with metamaterials. Nature Photon, 1: 224-227. doi: 10.1038/nphoton.2007.28 [32] Carrara M, Cacan M R, Leamy M J, Ruzzene M, Erturk A. 2012. Dramatic enhancement of structure-borne wave energy harvesting using an elliptical acoustic mirror. Applied Physics Letters, 100: 204105. doi: 10.1063/1.4719098 [33] Carrara M, Cacan M R, Toussaint J, Leamy M J, Ruzzene M, Erturk A. 2013. Metamaterial-inspired structures and concepts for elastoacoustic wave energy harvesting. Smart Mater. Struct., 22: 065004. doi: 10.1088/0964-1726/22/6/065004 [34] Chandra Y, Adhikari S, Mukherjee S, Mukhopadhyay T. 2022. Unfolding the mechanical properties of buckypaper composites: Nano-to-macro-scale coupled atomistic-continuum simulations. Engineering with Computers, 38: 5199-5229. doi: 10.1007/s00366-021-01538-w [35] Chandra Y, Mukhopadhyay T, Adhikari S, Figiel Ł. 2020. Size-dependent dynamic characteristics of graphene based multi-layer nano hetero-structures. Nanotechnology, 31: 145705. doi: 10.1088/1361-6528/ab6231 [36] Chaurha A, Malaji P V, Mukhopadhyay T. 2022. Dual functionality of vibration attenuation and energy harvesting: Effect of gradation on non-linear multi-resonator metastructures. Eur. Phys. J. Spec. Top., 231: 1403-1413. doi: 10.1140/epjs/s11734-022-00506-9 [37] Che K, Yuan C, Wu J, Jerry Qi H, Meaud J. 2016. Three-dimensional-printed multistable mechanical metamaterials with a deterministic deformation sequence. Journal of Applied Mechanics, 84 : 011004. [38] Chen B G, Upadhyaya N, Vitelli V. 2014. Nonlinear conduction via solitons in a topological mechanical insulator. Proceedings of the National Academy of Sciences, 111: 13004-13009. doi: 10.1073/pnas.1405969111 [39] Chen C, Lu T J, Fleck N A. 1999. Effect of imperfections on the yielding of two-dimensional foams. Journal of the Mechanics and Physics of Solids, 47: 2235-2272. doi: 10.1016/S0022-5096(99)00030-7 [40] Chen D, Zheng X. 2018. Multi-material additive manufacturing of metamaterials with giant, tailorable negative poisson’s ratios. Sci. Rep., 8: 9139. doi: 10.1038/s41598-018-26980-7 [41] Chen J S, Su W J, Cheng Y, Li W C, Lin C Y. 2019. A metamaterial structure capable of wave attenuation and concurrent energy harvesting. Journal of Intelligent Material Systems and Structures, 30: 2973-2981. doi: 10.1177/1045389X19880023 [42] Chen S, Cao Y, Sarparast M, Yuan H, Dong L, Tan X, Cao C. 2020. Soft crawling robots: Design, actuation, and locomotion. Advanced Materials Technologies, 5: 1900837. doi: 10.1002/admt.201900837 [43] Chen Y, Ai B, Wong Z J. 2020. Soft optical metamaterials. Nano Convergence, 7: 1-17. doi: 10.1186/s40580-019-0212-3 [44] Chen Y, Jin L. 2018. Geometric role in designing pneumatically actuated pattern-transforming metamaterials. Extreme Mechanics Letters, 23: 55-66. doi: 10.1016/j.eml.2018.08.001 [45] Chen Y, Ma Y, Yin Q, Pan F, Cui C, Zhang Z, Liu B. 2021. Advances in mechanics of hierarchical composite materials. Composites Science and Technology, 214: 108970. doi: 10.1016/j.compscitech.2021.108970 [46] Chen Y, Mai Y W, Ye L. 2023. Perspectives for multiphase mechanical metamaterials. Materials Science and Engineering: R: Reports, 153: 100725. doi: 10.1016/j.mser.2023.100725 [47] Cheng Y C, Lu H C, Lee X, Zeng H, Priimagi A. 2020. Kirigami-based light-induced shape-morphing and locomotion. Advanced Materials, 32: 1906233. doi: 10.1002/adma.201906233 [48] Cho Y, Shin J H, Costa A, Kim T A, Kunin V, Li J, Lee S Y, Yang S, Han H N, Choi I S, Srolovitz D J. 2014. Engineering the shape and structure of materials by fractal cut. Proceedings of the National Academy of Sciences, 111: 17390-17395. doi: 10.1073/pnas.1417276111 [49] Choi G P T, Dudte L H, Mahadevan L. 2019. Programming shape using kirigami tessellations. Nat. Mater., 18: 999-1004. doi: 10.1038/s41563-019-0452-y [50] Claeys C, Pluymers B, Sas P, Desmet W. 2014. Design of a resonant metamaterial based acoustic enclosure. Proceedings of the 26th International Conference on Noise and Vibration Engineering, 325 . [51] Compton B G, Lewis J A. 2014. 3d-printing of lightweight cellular composites. Advanced materials, 26: 5930-5935. doi: 10.1002/adma.201401804 [52] Cummer S A, Christensen J, Alù A. 2016. Controlling sound with acoustic metamaterials. Nat Rev Mater., 1: 1-13. [53] Cummer S A, Schurig D. 2007. One path to acoustic cloaking. New J. Phys., 9: 45. doi: 10.1088/1367-2630/9/3/045 [54] Cundy H M, Rollett A P. 1961. Mathematical Models. Oxford: Oxford University Press. [55] de Bruijn N G. 1981. Algebraic theory of penrose’s non-periodic tilings of the plane, kon. Nederl. Akad. Wetensch. Proc, Ser, 84: 1-7. doi: 10.1016/1385-7258(81)90013-5 [56] de Loos M, Feringa B L, van Esch J H. 2005. Design and application of self-assembled low molecular weight hydrogels. European Journal of Organic Chemistry, 17: 3615-3631. [57] de Moura B, Machado M R, Mukhopadhyay T, Dey S. 2022. Dynamic and wave propagation analysis of periodic smart beams coupled with resonant shunt circuits: Passive property modulation. Eur. Phys. J. Spec. Top., 231: 1415-1431. doi: 10.1140/epjs/s11734-022-00504-x [58] Del Vescovo D, Giorgio I. 2014. Dynamic problems for metamaterials: Review of existing models and ideas for further research. International Journal of Engineering Science, 80: 153-172. doi: 10.1016/j.ijengsci.2014.02.022 [59] dell’Isola F, Steigmann D, Corte A D. 2016. Synthesis of fibrous complex structures: Designing microstructure to deliver targeted macroscale response. Applied Mechanics Reviews, 67: 060804. [60] Demaine E, O’Rourke J. 2007. Geometric folding algorithms: linkages, origami, polyhedra, Cambridge university press. [61] Deshpande V S, Ashby M F, Fleck N A. 2001. Foam topology: Bending versus stretching dominated architectures. Acta Materialia, 49: 1035-1040. doi: 10.1016/S1359-6454(00)00379-7 [62] Dey S, Mukhopadhyay T, Adhikari S. 2018. Uncertainty quantification in laminated composites: A meta-model based approach. CRC Press. [63] Deymier P A. 2013. Acoustic metamaterials and phononic crystals. Springer science & Business media, 173 . [64] Dima A, Bhaskarla S, Becker C, Brady M, Campbell C, Dessauw P, Hanisch R, Kattner U, Kroenlein K, Newrock M, Peskin A, Plante R, Li S Y, Rigodiat P F, Amaral G S, Trautt Z, Schmitt X, Warren J, Youssef S. 2016. Informatics infrastructure for the materials genome initiative. JOM, 68: 2053-2064. doi: 10.1007/s11837-016-2000-4 [65] Ding Z, Yuan C, Peng X, Wang T, Qi H J, Dunn M L. 2017. Direct 4d printing via active composite materials. Science Advances, 3: e1602890. doi: 10.1126/sciadv.1602890 [66] Dudek K K, Gatt R, Dudek M R, Grima J N. 2018. Negative and positive stiffness in auxetic magneto-mechanical metamaterials. Proceedings of the Royal Society A: Mathematical. Physical and Engineering Sciences, 474: 20180003. doi: 10.1098/rspa.2018.0003 [67] Dudek K K, Gatt R, Grima J N. 2020. 3D composite metamaterial with magnetic inclusions exhibiting negative stiffness and auxetic behaviour. Materials & Design, 187: 108403. [68] Dudek K K, Martínez J A I, Ulliac G, Kadic M. 2022. Micro-scale auxetic hierarchical mechanical metamaterials for shape morphing. Advanced Materials, 34: 2110115. doi: 10.1002/adma.202110115 [69] Dudek K K, Wolak W, Gatt R, Grima J N. 2019. Impact resistance of composite magnetic metamaterials. Sci. Rep., 9: 3963. doi: 10.1038/s41598-019-40610-w [70] Dudek M R, Dudek K K, Wolak W, Wojciechowski K W, Grima J N. 2019. Magnetocaloric materials with ultra-small magnetic nanoparticles working at room temperature. Sci. Rep., 9:17607. doi: 10.1038/s41598-018-37186-2 [71] Dudte L H, Vouga E, Tachi T, Mahadevan L. 2016. Programming curvature using origami tessellations. Nature Mater., 15: 583-588. doi: 10.1038/nmat4540 [72] Eidini M, Paulino G H. 2015. Unraveling metamaterial properties in zigzag-base folded sheets. Science Advances, 1: e1500224. doi: 10.1126/sciadv.1500224 [73] El Helou C, Buskohl P R, Tabor C E, Harne R L. 2021. Digital logic gates in soft, conductive mechanical metamaterials. Nat. Commun., 12: 1-8. doi: 10.1038/s41467-020-20314-w [74] Evans A A, Silverberg J L, Santangelo C D. 2015. Lattice mechanics of origami tessellations. Phys. Rev. E, 92: 013205. doi: 10.1103/PhysRevE.92.013205 [75] Evans K E, Alderson A. 2000. Auxetic materials: Functional materials and structures from lateral thinking! Advanced Materials, 12 : 617–628. [76] Fahy F, Walker J. 2018. Advanced Applications in Acoustics, Noise and Vibration. Boca Raton: CRC Press. [77] Fan H, Yang L, Tian Y, Wang Z. 2020. Design of metastructures with quasi-zero dynamic stiffness for vibration isolation. Composite Structures, 243: 112244. doi: 10.1016/j.compstruct.2020.112244 [78] Fan Y, Collet M, Ichchou M, Li L, Bareille O, Dimitrijevic Z. 2017. Enhanced wave and finite element method for wave propagation and forced response prediction in periodic piezoelectric structures. Chinese Journal of Aeronautics, 30: 75-87. doi: 10.1016/j.cja.2016.12.011 [79] Fang H, Chu S C A, Xia Y, Wang K W. 2018. Programmable self-locking origami mechanical metamaterials. Advanced Materials, 30: 1706311. doi: 10.1002/adma.201706311 [80] Fang H, Li S, Ji H, Wang K W. 2016. Uncovering the deformation mechanisms of origami metamaterials by introducing generic degree-four vertices. Phys. Rev. E, 94: 043002. doi: 10.1103/PhysRevE.94.043002 [81] Fang N, Lee H, Sun C, Zhang X. 2005. Sub-diffraction-limited optical imaging with a silver superlens. Science, 308: 534-537. doi: 10.1126/science.1108759 [82] Fang Z, Song H, Zhang Y, Jin B, Wu J, Zhao Q, Xie T. 2020. Modular 4d printing via interfacial welding of digital light-controllable dynamic covalent polymer networks. Matter., 2: 1187-1197. doi: 10.1016/j.matt.2020.01.014 [83] Fathers R K, Gattas J M, You Z. 2015. Quasi-static crushing of eggbox, cube, and modified cube foldcore sandwich structures. International Journal of Mechanical Sciences, 101: 421-428. [84] Filipov E T, Tachi T, Paulino G H. 2015. Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proceedings of the National Academy of Sciences, 112: 12321-12326. doi: 10.1073/pnas.1509465112 [85] Fleck N A, Deshpande V S, Ashby M F. 2010. Micro-architectured materials: Past, present and future. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466: 2495-2516. doi: 10.1098/rspa.2010.0215 [86] Florijn B, Coulais C, van Hecke M. 2014. Programmable mechanical metamaterials. Phys. Rev. Lett., 113: 175503. doi: 10.1103/PhysRevLett.113.175503 [87] Frenzel T, Kadic M, Wegener M. 2017. Three-dimensional mechanical metamaterials with a twist. Science, 358: 1072-1074. doi: 10.1126/science.aao4640 [88] Galea R, Dudek K K, Farrugia P S, Zammit Mangion L, Grima J N, Gatt R. 2022. Reconfigurable magneto-mechanical metamaterials guided by magnetic fields. Composite Structures, 280: 114921. doi: 10.1016/j.compstruct.2021.114921 [89] Gao H, Ji B, Jäger I L, Arzt E, Fratzl P. 2003. Materials become insensitive to flaws at nanoscale: Lessons from nature. Proceedings of the National Academy of Sciences, 100: 5597-5600. doi: 10.1073/pnas.0631609100 [90] Gardner D F, Evans J S, Smalyukh I I. 2011. Towards reconfigurable optical metamaterials: Colloidal nanoparticle self-assembly and self-alignment in liquid crystals. Molecular Crystals and Liquid Crystals, 545: 3-1227. [91] Gatt R, Grima J N. 2008. Negative compressibility. Physica Status Solidi, 2: 236-238. [92] Gatt R, Mizzi L, Azzopardi J I, Azzopardi K M, Attard D, Casha A, Briffa J, Grima J N. 2015. Hierarchical auxetic mechanical metamaterials. Sci. Rep., 5: 8395. doi: 10.1038/srep08395 [93] Ge Q, Dunn C K, Qi H J, Dunn M L. 2014. Active origami by 4D printing. Smart Mater. Struct., 23: 094007. doi: 10.1088/0964-1726/23/9/094007 [94] Ge Q, Qi H J, Dunn M L. 2013. Active materials by four-dimension printing. Applied Physics Letters, 103: 131901. doi: 10.1063/1.4819837 [95] Ghuku S, Mukhopadhyay T. 2023. On enhancing mode-dependent failure strength under large deformation: The concept of anti-curvature in honeycomb lattices. Composite Structures, 305: 116318. doi: 10.1016/j.compstruct.2022.116318 [96] Ghuku S, Mukhopadhyay T. 2022. Anti-curvature honeycomb lattices for mode-dependent enhancement of nonlinear elastic properties under large deformation. International Journal of Non-Linear Mechanics, 140: 103887. doi: 10.1016/j.ijnonlinmec.2021.103887 [97] Gibson I J, Ashby M F. 1997. The mechanics of three-dimensional cellular materials. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 382: 43-59. [98] Gibson L J. 2012. The hierarchical structure and mechanics of plant materials. Journal of The Royal Society Interface, 9: 2749-2766. doi: 10.1098/rsif.2012.0341 [99] Gibson L J, Ashby M F, Schajer G S, Robertson, C I. 1997. The mechanics of two-dimensional cellular materials. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 382: 25-42. [100] Giri T R, Mailen R. 2021. Controlled snapping sequence and energy absorption in multistable mechanical metamaterial cylinders. International Journal of Mechanical Sciences, 204: 106541. doi: 10.1016/j.ijmecsci.2021.106541 [101] Gleeson H. 2020. Stretching the limits. Phys. World, 33: 36. [102] Goldsmith B R, Esterhuizen J, Liu J X, Bartel C J, Sutton C. 2018. Machine learning for heterogeneous catalyst design and discovery. AIChE Journal, 64: 2311-2323. doi: 10.1002/aic.16198 [103] Gómez L R, Turner A M, van Hecke M, Vitelli V. 2012. Shocks near Jamming. Phys. Rev. Lett., 108: 058001. doi: 10.1103/PhysRevLett.108.058001 [104] Greer J R, Deshpande V S. 2019. Three-dimensional architected materials and structures: Design, fabrication, and mechanical behavior. MRS Bulletin, 44: 750-757. doi: 10.1557/mrs.2019.232 [105] Grima J N, Caruana-Gauci R, Dudek M R, Wojciechowski K W, Gatt, R. 2013. Smart metamaterials with tunable auxetic and other properties. Smart Mater. Struct., 22: 084016. doi: 10.1088/0964-1726/22/8/084016 [106] Grimberg R. 2013. Electromagnetic metamaterials. Materials Science and Engineering, 178: 1285-1295. doi: 10.1016/j.mseb.2013.03.022 [107] Grosskopf A K, Truby R L, Kim H, Perazzo A, Lewis J A, Stone H A. 2018. Viscoplastic matrix materials for embedded 3d printing. ACS Appl. Mater. Interfaces, 10: 23353-23361. doi: 10.1021/acsami.7b19818 [108] Gu G X, Chen C T, Buehler M J. 2018a. De novo composite design based on machine learning algorithm. Extreme Mechanics Letters, 18: 19-28. doi: 10.1016/j.eml.2017.10.001 [109] Gu G X, Chen C T, Richmond D J, Buehler M J. 2018b. Bioinspired hierarchical composite design using machine learning: Simulation, additive manufacturing, and experiment. Mater. Horiz., 5: 939-945. doi: 10.1039/C8MH00653A [110] Gu G X, Wettermark, S, Buehler, M J. 2017. Algorithm-driven design of fracture resistant composite materials realized through additive manufacturing. Additive Manufacturing, 17: 47-54. doi: 10.1016/j.addma.2017.07.002 [111] Gu H, Boehler Q, Cui H, Secchi E, Savorana G, De Marco C, Gervasoni S, Peyron Q, Huang T Y, Pane S, Hirt A M, Ahmed D, Nelson B J. 2020. Magnetic cilia carpets with programmable metachronal waves. Nat. Commun., 11: 2637. doi: 10.1038/s41467-020-16458-4 [112] Guenneau S, Movchan A, Pétursson G, Ramakrishna S A. 2007. Acoustic metamaterials for sound focusing and confinement. New J. Phys., 9: 399. doi: 10.1088/1367-2630/9/11/399 [113] Guo K, Yang Z, Yu C H, Buehler M J. 2021. Artificial intelligence and machine learning in design of mechanical materials. Mater. Horiz., 8: 1153-1172. doi: 10.1039/D0MH01451F [114] Gupta K, Mukhopadhyay T, Roy L, Dey S. 2022. High-velocity ballistics of twisted bilayer graphene under stochastic disorder. Adv. Nano Res, 12: 529-547. [115] Gupta K K, Roy A, Mukhopadhyay T, Roy L, Dey S. 2022. Probing the stochastic fracture behavior of twisted bilayer graphene: Efficient ann based molecular dynamics simulations for complete probabilistic characterization. Materials Today Communications, 32: 103932. doi: 10.1016/j.mtcomm.2022.103932 [116] Gupta S, Mukhopadhyay T, Kushvaha V. 2023. Microstructural image based convolutional neural networks for efficient prediction of full-field stress maps in short fiber polymer composites. Defence Technology, 24: 58-82. doi: 10.1016/j.dt.2022.09.008 [117] Ha C S, Lakes R S, Plesha M E. 2018. Design, fabrication, and analysis of lattice exhibiting energy absorption via snap-through behavior. Materials & Design, 141: 426-437. [118] Hahn V, Kiefer P, Frenzel T, Qu J, Blasco E, Barner-Kowollik C, Wegener M. 2020. Rapid assembly of small materials building blocks (voxels) into large functional 3d metamaterials. Advanced Functional Materials, 30: 1907795. doi: 10.1002/adfm.201907795 [119] Hao X P, Xu Z, Li C Y, Hong W, Zheng Q, Wu Z L. 2020. Kirigami-design-enabled hydrogel multimorphs with application as a multistate switch. Advanced Materials, 32: 2000781. doi: 10.1002/adma.202000781 [120] He L, Wen Z, Jin Y, Torrent D, Zhuang X, Rabczuk T. 2021. Inverse design of topological metaplates for flexural waves with machine learning. Materials & Design, 199: 109390. [121] Hedayati R, Leeflang A M, Zadpoor A A. 2017. Additively manufactured metallic pentamode meta-materials. Applied Physics Letters, 110: 091905. doi: 10.1063/1.4977561 [122] Helma C, Cramer T, Kramer S, De Raedt L. 2004. Data mining and machine learning techniques for the identification of mutagenicity inducing substructures and structure activity relationships of noncongeneric compounds. J. Chem. Inf. Comput. Sci, 44: 1402-1411. doi: 10.1021/ci034254q [123] Hewage T A M, Alderson K L, Alderson A, Scarpa F. 2016. Double-negative mechanical metamaterials displaying simultaneous negative stiffness and negative poisson’s ratio properties. Advanced Materials, 28: 10323-10332. doi: 10.1002/adma.201603959 [124] Hu N, Burgueño R. 2015. Buckling-induced smart applications: Recent advances and trends. Smart Mater. Struct, 24: 063001. doi: 10.1088/0964-1726/24/6/063001 [125] Hua J, Lei H, Gao C F, Guo X, Fang D. 2020. Parameters analysis and optimization of a typical multistable mechanical metamaterial. Extreme Mechanics Letters, 35: 100640. doi: 10.1016/j.eml.2020.100640 [126] Huang C, Chen L. 2016. Negative poisson’s ratio in modern functional materials. Advanced Materials, 28: 8079-8096. doi: 10.1002/adma.201601363 [127] Huang J, Zhang J, Xu D, Zhang S, Tong H, Xu N. 2023. From jammed solids to mechanical metamaterials : A brief review. Current Opinion in Solid State and Materials Science, 27: 101053. doi: 10.1016/j.cossms.2022.101053 [128] Hussein M I. 2009. Reduced bloch mode expansion for periodic media band structure calculations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465: 2825-2848. doi: 10.1098/rspa.2008.0471 [129] Hussein M I, Frazier M J. 2013. Metadamping: An emergent phenomenon in dissipative metamaterials. Journal of Sound and Vibration, 332: 4767-4774. doi: 10.1016/j.jsv.2013.04.041 [130] Hussein M I, Leamy M J, Ruzzene M. 2014. Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook. Applied Mechanics Reviews, 66 : 040802. [131] Imre A R. 2014. Metamaterials with negative compressibility—A novel concept with a long history. Mater. Sci. -Pol., 32: 126-129. doi: 10.2478/s13536-013-0179-4 [132] Inman D J. 2017. Vibration with control. John Wiley & Sons. [133] Isanaka B R, Mukhopadhyay T, Varma R K, Kushvaha V. 2022. On exploiting machine learning for failure pattern driven strength enhancement of honeycomb lattices. Acta Materialia, 239: 118226. doi: 10.1016/j.actamat.2022.118226 [134] Isobe M, Okumura K. 2016. Initial rigid response and softening transition of highly stretchable kirigami sheet materials. Sci. Rep., 6: 24758. doi: 10.1038/srep24758 [135] Jackson J A, Messner M C, Dudukovic N A, Smith W L, Bekker L, Moran B, Golobic A M, Pascall A J, Duoss E B, Loh K J, Spadaccini C M. 2018. Field responsive mechanical metamaterials. Science Advances, 4: eaau6419. doi: 10.1126/sciadv.aau6419 [136] Jacob Z, Alekseyev L V, Narimanov E. 2006. Optical hyperlens: Far-field imaging beyond the diffraction limit. Opt. Express, 14: 8247. doi: 10.1364/OE.14.008247 [137] Jain A, Ong S P, Hautier G, Chen W, Richards W D, Dacek S, Cholia S, Gunter D, Skinner D, Ceder G, Persson K A. 2013. Commentary: The materials project: A materials genome approach to accelerating materials innovation. APL Materials, 1: 011002. doi: 10.1063/1.4812323 [138] Jakšić Z, Jakšić O, Djurić Z, Kment C. 2007. A consideration of the use of metamaterials for sensing applications: Field fluctuations and ultimate performance. J. Opt. A: Pure Appl. Opt., 9: S377. doi: 10.1088/1464-4258/9/9/S16 [139] Janbaz S, Bobbert F S L, Mirzaali M J, Zadpoor A A. 2019. Ultra-programmable buckling-driven soft cellular mechanisms. Mater. Horiz., 6: 1138-1147. doi: 10.1039/C9MH00125E [140] Janbaz S, Noordzij N, Widyaratih D S, Hagen C W, Fratila-Apachitei L E, Zadpoor A A. 2017. Origami lattices with free-form surface ornaments. Science Advances, 3: eaao1595. doi: 10.1126/sciadv.aao1595 [141] Jang D, Greer J R. 2010. Transition from a strong-yet-brittle to a stronger-and-ductile state by size reduction of metallic glasses. Nature Mater., 9: 215-219. doi: 10.1038/nmat2622 [142] Jena D P, Panigrahi S N, Kumar R. 2013. Gear fault identification and localization using analytic wavelet transform of vibration signal. Measurement, 46: 1115-1124. doi: 10.1016/j.measurement.2012.11.010 [143] Jenett B, Cameron C, Tourlomousis F, Rubio A P, Ochalek M, Gershenfeld N. 2020. Discretely assembled mechanical metamaterials. Science Advances, 6: eabc9943. doi: 10.1126/sciadv.abc9943 [144] Ji X, Deng L, Zhang J, Luan Y, Duan Y. 2022. Energy absorption characteristics of 3d lattice structure filled with periodic inner core based on 3d printing. J. of Materi Eng and Perform, 31: 6784-6794. doi: 10.1007/s11665-022-06692-w [145] Jia H, Gu S Y, Chang K. 2018. 3D printed self-expandable vascular stents from biodegradable shape memory polymer. Advances in Polymer Technology, 37: 3222-3228. doi: 10.1002/adv.22091 [146] Jiang C, Rist F, Wang H, Wallner J, Pottmann H. 2022. Shape-morphing mechanical metamaterials. Computer-Aided Design, 143: 103146. doi: 10.1016/j.cad.2021.103146 [147] Jikov V V, Kozlov S M, Oleinik O A. 1994. Homogenization of differential operators and integral functionals. Springer Science & Business Media. [148] Jin L, Forte A E, Deng B, Rafsanjani A, Bertoldi K. 2020. Kirigami-inspired inflatables with programmable shapes. Advanced Materials, 32: 2001863. doi: 10.1002/adma.202001863 [149] Jin Y, He L, Wen Z, Mortazavi B, Hongwei G, Torrent D, Djafari-Rouhani B, Rabczuk T, Zhuang X, Li Y. 2022. Intelligent on-demand design of phononic metamaterials. Nanophotonics, 11: 439-460. doi: 10.1515/nanoph-2021-0639 [150] Jones D R H, Ashby M F. 2011. Engineering Materials 1: An Introduction to Properties, Application and Design. Oxford: Elsevier. [151] Kadic M, Bückmann T, Stenger N, Thiel M, Wegener M. 2012. On the practicability of pentamode mechanical metamaterials. Applied Physics Letters, 100: 191901. doi: 10.1063/1.4709436 [152] Kadic M, Milton G W, van Hecke M, Wegener M. 2019. 3d metamaterials. Nat Rev Phys, 1: 198-210. doi: 10.1038/s42254-018-0018-y [153] Kaur M, Yun T G, Han S M, Thomas E L, Kim W S. 2017. 3D printed stretching-dominated micro-trusses. Materials & Design, 134: 272-280. [154] Kim D, Ferretto I, Leinenbach C, Lee W. 2022. 3d and 4d printing of complex structures of Fe-mn-Si-based shape memory alloy using laser powder bed fusion. Advanced Materials Interfaces, 9: 2200171. doi: 10.1002/admi.202200171 [155] Kim W, Byun J, Kim J K, Choi W Y, Jakobsen K, Jakobsen J, Lee D Y, Cho K J. 2019. Bioinspired dual-morphing stretchable origami. Science Robotics, 4: eaay3493. doi: 10.1126/scirobotics.aay3493 [156] Kim Y, Parada G A, Liu S, Zhao X. 2019. Ferromagnetic soft continuum robots. Science Robotics, 4: eaax7329. doi: 10.1126/scirobotics.aax7329 [157] Kim Y, Yuk H, Zhao R, Chester S A, Zhao X. 2018. Printing ferromagnetic domains for untethered fast-transforming soft materials. Nature, 558: 274-279. doi: 10.1038/s41586-018-0185-0 [158] Kirklin S, Saal J E, Meredig B, Thompson A, Doak J W, Aykol M, Rühl S, Wolverton C. 2015. The open quantum materials database (oqmd): Assessing the accuracy of dft formation energies. npj Comput Mater., 1: 1-15. [159] Kokkinis D, Bouville F, Studart A R. 2018. 3d printing of materials with tunable failure via bioinspired mechanical gradients. Advanced Materials, 30: 1705808. doi: 10.1002/adma.201705808 [160] Kokkinis D, Schaffner M, Studart A R. 2015. Multimaterial magnetically assisted 3d printing of composite materials. Nat. Commun., 6: 8643. doi: 10.1038/ncomms9643 [161] Kozin V K, Shelykh I A, Nalitov A V, Iorsh I V. 2018. Topological metamaterials based on polariton rings. Phys. Rev. B, 98: 125115. doi: 10.1103/PhysRevB.98.125115 [162] Krishnamoorthy H N S, Jacob Z, Narimanov E, Kretzschmar I, Menon V M. 2012. Topological transitions in metamaterials. Science, 336: 205-209. doi: 10.1126/science.1219171 [163] Krödel S, Thomé N, Daraio C. 2015. Wide band-gap seismic metastructures. Extreme Mechanics Letters, 4: 111-117. doi: 10.1016/j.eml.2015.05.004 [164] Kruk S S, Wong Z J, Pshenay-Severin E, O’Brien K, Neshev D N, Kivshar Y S, Zhang X. 2016. Magnetic hyperbolic optical metamaterials. Nat. Commun., 7: 11329. doi: 10.1038/ncomms11329 [165] Kruth J P, Froyen L, Van Vaerenbergh J, Mercelis P, Rombouts M, Lauwers B. 2004. Selective laser melting of iron-based powder. Journal of Materials Processing Technology, 149: 616-622. doi: 10.1016/j.jmatprotec.2003.11.051 [166] Kuang X, Roach D J, Wu J, Hamel C M, Ding Z, Wang T, Dunn M L, Qi H J. 2019a. Advances in 4d printing: Materials and applications. Advanced Functional Materials, 29: 1805290. doi: 10.1002/adfm.201805290 [167] Kuang X, Wu J, Chen K, Zhao Z, Ding Z, Hu F, Fang D, Qi H J. 2019. Grayscale digital light processing 3D printing for highly functionally graded materials. Science Advances, 5: eaav5790. doi: 10.1126/sciadv.aav5790 [168] Kundu D, Ghuku S, Naskar S, Mukhopadhyay T. 2023. Extreme specific stiffness through interactive cellular networks in bi-level micro-topology architected metamaterials. Advanced Engineering Materials, 25: 2201407. doi: 10.1002/adem.202201407 [169] Kuribayashi K, Tsuchiya K, You Z, Tomus D, Umemoto M, Ito T, Sasaki M. 2006. Self-deployable origami stent grafts as a biomedical application of Ni-rich TiNi shape memory alloy foil. Materials Science and Engineering: A, 419: 131-137. doi: 10.1016/j.msea.2005.12.016 [170] Lakes R. 1993. Advances in negative Poisson’s ratio materials. Advanced Materials, 5: 293-296. doi: 10.1002/adma.19930050416 [171] Lakes R. 1987. Foam structures with a negative Poisson’s ratio. Science, 235: 1038-1040. doi: 10.1126/science.235.4792.1038 [172] Lakes R, Wojciechowski K W. 2008. Negative compressibility, negative Poisson’s ratio, and stability. Physica Status Solidi, 245: 545-551. doi: 10.1002/pssb.200777708 [173] Lakes R S, Lee T, Bersie A, Wang Y C. 2001. Extreme damping in composite materials with negative-stiffness inclusions. Nature, 410: 565-567. doi: 10.1038/35069035 [174] Lamoureux A, Lee K, Shlian M, Forrest S R, Shtein M. 2015. Dynamic kirigami structures for integrated solar tracking. Nat. Commun., 6: 8092. doi: 10.1038/ncomms9092 [175] Lang J P, Jiang W, Teng X C, Zhang X G, Han D, Hao J, Xu H H, Ni X H, Xie Y M, Qin Q H, Yang J, Ren X. 2023. Assembled mechanical metamaterials with transformable shape and auxeticity. Construction and Building Materials, 378: 131181. doi: 10.1016/j.conbuildmat.2023.131181 [176] Lang R J. 2012. Origami design secrets: Mathematical methods for an ancient art. CRC Press. [177] Lee D Y, Kim J S, Kim S R, Koh J S, Cho K J. 2014. The deformable wheel robot using magic-ball origami structure. International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 55942. [178] Lee H J, Yook J G. 2008. Biosensing using split-ring resonators at microwave regime. Applied Physics Letters, 92: 254103. doi: 10.1063/1.2946656 [179] Lee J H, Singer J P, Thomas E L. 2012. Micro-/nanostructured mechanical metamaterials. Advanced Materials, 24: 4782-4810. doi: 10.1002/adma.201201644 [180] Lee N, Yoon B, Kim T, Bae J Y, Lim J S, Chang I, Cho H H. 2020. Multiple resonance metamaterial emitter for deception of infrared emission with enhanced energy dissipation. ACS Appl. Mater. Interfaces, 12: 8862-8869. doi: 10.1021/acsami.9b21030 [181] Lee S H, Park C M, Seo Y M, Wang Z G, Kim C K. 2009a. Acoustic metamaterial with negative density. Physics Letters A, 373: 4464-4469. doi: 10.1016/j.physleta.2009.10.013 [182] Lee S H, Park C M, Seo Y M, Wang Z G, Kim C K. 2009b. Acoustic metamaterial with negative modulus. J. Phys: Condens. Matter, 21: 175704. doi: 10.1088/0953-8984/21/17/175704 [183] Lei M, Hamel C M, Yuan C, Lu H, Qi H J. 2018. 3d printed two-dimensional periodic structures with tailored in-plane dynamic responses and fracture behaviors. Composites Science and Technology, 159: 189-198. doi: 10.1016/j.compscitech.2018.02.024 [184] Lei M, Hong W, Zhao Z, Hamel C, Chen M, Lu H, Qi H J. 2019. 3d printing of auxetic metamaterials with digitally reprogrammable shape. ACS Appl. Mater. Interfaces, 11: 22768-22776. doi: 10.1021/acsami.9b06081 [185] Levine D J, Turner K T, Pikul J H. 2021. Materials with electroprogrammable stiffness. Advanced Materials, 33: 2007952. doi: 10.1002/adma.202007952 [186] Li J, Chan C T. 2004. Double-negative acoustic metamaterial. Phys. Rev. E, 70: 055602. doi: 10.1103/PhysRevE.70.055602 [187] Li K, Cheng X, Zhu F, Li L Z, Xie Z, Luan H, Wang Z, Ji Z, Wang H, Liu F, Xue Y, Jiang C, Feng X, Li L M, Rogers J A, Huang Y, Zhang Y. 2019. A generic soft encapsulation strategy for stretchable electronics. Advanced Functional Materials, 29: 1806630. doi: 10.1002/adfm.201806630 [188] Li S, Stampfli J J, Xu H J, Malkin E, Diaz E V, Rus D, Wood R J. 2019. A vacuum-driven origami “magic-ball” soft gripper. 2019 International Conference on Robotics and Automation (ICRA), 7401–7408. [189] Li S, Wang K W. 2015. Fluidic origami: A plant-inspired adaptive structure with shape morphing and stiffness tuning. Smart Mater. Struct., 24: 105031. doi: 10.1088/0964-1726/24/10/105031 [190] Li T, Wang J, Zhang L, Yang J, Yang M, Zhu D, Zhou X H, Handschuh-Wang S, Liu Y, Zhou X C. 2017. “Freezing”, morphing, and folding of stretchy tough hydrogels. J. Mater. Chem. B, 5: 5726-5732. doi: 10.1039/C7TB01265A [191] Li W, Matsuhisa N, Liu Z Y, Wang M, Luo Y, Cai P, Chen G, Zhang F, Li C, Liu Z H, Lv Z, Zhang W, Chen X. 2021. An on-demand plant-based actuator created using conformable electrodes. Nat. Electron., 4: 134-142. doi: 10.1038/s41928-020-00530-4 [192] Libonati F, Buehler M J. 2017. Advanced structural materials by bioinspiration. Advanced Engineering Materials, 19: 1600787. doi: 10.1002/adem.201600787 [193] Lier E, Shaw R K. 2008. Design and simulation of metamaterial-based hybrid-mode horn antennas. Electronics Letters, 44: 1444-1445. doi: 10.1049/el:20082639 [194] Lier E, Werner D H, Scarborough C P, Wu Q, Bossard J A. 2011. An octave-bandwidth negligible-loss radiofrequency metamaterial. Nat. Mater., 10: 216-222. doi: 10.1038/nmat2950 [195] Ligon S C, Liska R, Stampfl J, Gurr M, Mülhaupt R. 2017. Polymers for 3d printing and customized additive manufacturing. Chem. Rev., 117: 10212-10290. doi: 10.1021/acs.chemrev.7b00074 [196] Lim T C. 2015. Auxetic materials and structures. Engineering Materials. [197] Liu J, Gu T, Shan S, Kang S H, Weaver J C, Bertoldi K. 2016. Harnessing buckling to design architected materials that exhibit effective negative swelling. Advanced Materials, 28: 6619-6624. doi: 10.1002/adma.201600812 [198] Liu Q, Wang W, Reynolds M F, Cao M C, Miskin M Z, Arias T A, Muller D A, McEuen P L, Cohen I. 2021. Micrometer-sized electrically programmable shape-memory actuators for low-power microrobotics. Science Robotics, 6: eabe6663. doi: 10.1126/scirobotics.abe6663 [199] Liu R, Yabansu Y C, Yang Z, Choudhary A N, Kalidindi S R, Agrawal A. 2017. Context aware machine learning approaches for modeling elastic localization in three-dimensional composite microstructures. Integr Mater Manuf Innov, 6: 160-171. doi: 10.1007/s40192-017-0094-3 [200] Liu S, Azad A I, Burgueño R. 2019. Architected materials for tailorable shear behavior with energy dissipation. Extreme Mechanics Letters, 28: 1-7 doi: 10.1016/j.eml.2019.01.010 [201] Liu X N, Hu G K, Huang G L, Sun C T. 2011. An elastic metamaterial with simultaneously negative mass density and bulk modulus. Applied Physics Letters, 98: 251907. doi: 10.1063/1.3597651 [202] Liu Y, Shaw B, Dickey M D, Genzer J. 2017. Sequential self-folding of polymer sheets. Science Advances, 3: e1602417. doi: 10.1126/sciadv.1602417 [203] Liu Y, Wang H, Ho J, Ng R C, Ng R J H, Hall-Chen V H, Koay E H H, Dong Z, Liu H, Qiu C W, Greer J R, Yang J K W. 2019. Structural color three-dimensional printing by shrinking photonic crystals. Nat. Commun., 10: 4340. doi: 10.1038/s41467-019-12360-w [204] Liu Z, Du H, Li J, Lu L, Li Z Y, Fang N X. 2018. Nano-kirigami with giant optical chirality. Science Advances, 4: eaat4436. doi: 10.1126/sciadv.aat4436 [205] Liu Z, Zhang X, Mao Y, Zhu Y Y, Yang Z, Chan C T, Sheng P. 2000. Locally resonant sonic materials. Science, 289: 1734-1736. doi: 10.1126/science.289.5485.1734 [206] Lockwood E H, MacMillan R H, Geometric Symmetry. cambridge university press, 1978. [207] Lorna J Gibson, Michael F Ashby. 1999. Cellular solids structure and properties. Cambridge University Press. [208] Lu M H, Feng L, Chen Y F. 2009. Phononic crystals and acoustic metamaterials. Materials Today, 12: 34-42. [209] Lum G Z, Ye Z, Dong X, Marvi H, Erin O, Hu W, Sitti M. 2016. Shape-programmable magnetic soft matter. Proceedings of the National Academy of Sciences, 113(41): E6007-E6015. [210] Luo C, Ning S, Liu Z, Zhuang Z. 2020. Interactive inverse design of layered phononic crystals based on reinforcement learning. Extreme Mechanics Letters, 36: 100651. doi: 10.1016/j.eml.2020.100651 [211] Lv H, Tian X, Wang M Y, Li, D. 2013. Vibration energy harvesting using a phononic crystal with point defect states. Applied Physics Letters, 102: 034103. doi: 10.1063/1.4788810 [212] Ma G, Yang M, Xiao S, Yang Z, Sheng P. 2014. Acoustic metasurface with hybrid resonances. Nat. Mater., 13: 873-878. doi: 10.1038/nmat3994 [213] Ma H, Wang K, Zhao H, Shi W, Xue J, Zhou Y, Li Q, Wang G, Yan B. 2022. Energy dissipation and shock isolation using novel metamaterials. International Journal of Mechanical Sciences, 228: 107464. doi: 10.1016/j.ijmecsci.2022.107464 [214] Ma H S, Prévost J H, Jullien R, Scherer G W. 2001. Computer simulation of mechanical structure–property relationship of aerogels. Journal of Non-Crystalline Solids, 285: 216-221. doi: 10.1016/S0022-3093(01)00456-2 [215] Ma W, Cheng F, Liu Y. 2018. Deep-learning-enabled on-demand design of chiral metamaterials. ACS Nano, 12: 6326-6334. doi: 10.1021/acsnano.8b03569 [216] Machado M R, Moura B B, Dey S, Mukhopadhyay T. 2022. Bandgap manipulation of single and multi-frequency smart metastructures with random impedance disorder. Smart Mater. Struct., 31: 105020. doi: 10.1088/1361-665X/ac8ef9 [217] Mahata A, Mukhopadhyay T. 2018. Probing the chirality-dependent elastic properties and crack propagation behavior of single and bilayer stanene. Phys. Chem. Chem. Phys., 20: 22768-22782. doi: 10.1039/C8CP03892A [218] Malek S, Gibson L. 2015. Effective elastic properties of periodic hexagonal honeycombs. Mechanics of Materials, 91: 226-240. doi: 10.1016/j.mechmat.2015.07.008 [219] Mao M, He J, Li X, Zhang B, Lei Q, Liu Y, Li D. 2017. The emerging frontiers and applications of high-resolution 3d printing. Micromachines, 8: 113. doi: 10.3390/mi8040113 [220] Mao Y, He Q, Zhao X. 2020. Designing complex architectured materials with generative adversarial networks. Science Advances, 6: eaaz4169. doi: 10.1126/sciadv.aaz4169 [221] Mead D M. 1996. Wave propagation in continuous periodic structures: Research contributions from southampton, 1964–1995. Journal of Sound and Vibration, 190: 495-524. doi: 10.1006/jsvi.1996.0076 [222] Meeussen A S, Paulose J, Vitelli V. 2016. Geared topological metamaterials with tunable mechanical stability. Phys. Rev. X, 6: 041029. [223] Meza L R, Das S, Greer J R. 2014. Strong, lightweight, and recoverable three-dimensional ceramic nanolattices. Science, 345: 1322-1326. doi: 10.1126/science.1255908 [224] Milton G W. 2015. New examples of three-dimensional dilational materials. Physica Status Solidi, 252: 1426-1430. doi: 10.1002/pssb.201552297 [225] Milton G W, Cherkaev A V. 1995. Which elasticity tensors are realizable. Journal of Engineering Materials and Technology, 117: 483-493. doi: 10.1115/1.2804743 [226] Mirzaali M J, Caracciolo A, Pahlavani H, Janbaz S, Vergani L, Zadpoor A A. 2018. Multi-material 3d printed mechanical metamaterials: rational design of elastic properties through spatial distribution of hard and soft phases. Applied Physics Letters, 113: 241903. doi: 10.1063/1.5064864 [227] Mirzaali M J, Ghorbani A, Nakatani K, Nouri-Goushki M, Tümer N, Callens S J P, Janbaz S, Accardo A, Bico J, Habibi M, Zadpoor A A. 2021. Curvature induced by deflection in thick meta-plates. Advanced Materials, 33: 2008082. doi: 10.1002/adma.202008082 [228] Mishin Y. 2021. Machine-learning interatomic potentials for materials science. Acta Materialia, 214: 116980. doi: 10.1016/j.actamat.2021.116980 [229] Mishra A K, Wallin T J, Pan W, Xu A, Wang K, Giannelis E P, Mazzolai B, Shepherd R F. 2020. Autonomic perspiration in 3d-printed hydrogel actuators. Science Robotics, 5: eaaz3918. doi: 10.1126/scirobotics.aaz3918 [230] Montgomery S M, Wu S, Kuang X, Armstrong C D, Zemelka C, Ze Q, Zhang R, Zhao R, Qi H J. 2021. Magneto-mechanical metamaterials with widely tunable mechanical properties and acoustic bandgaps. Advanced Functional Materials, 31: 2005319. doi: 10.1002/adfm.202005319 [231] Mortazavi B, Zhuang X, Rabczuk T, Shapeev A V. 2023. Atomistic modeling of the mechanical properties: The rise of machine learning interatomic potentials. Mater. Horiz., 10: 1956-1968. doi: 10.1039/D3MH00125C [232] Moruzzi M C, Cinefra M, Bagassi S. 2021. Vibroacoustic analysis of an innovative windowless cabin with metamaterial trim panels in regional turboprops. Mechanics of Advanced Materials and Structures, 28: 1509-1521. doi: 10.1080/15376494.2019.1682729 [233] Mousanezhad D, Haghpanah B, Ghosh R, Hamouda A M, Nayeb-Hashemi H, Vaziri A. 2016. Elastic properties of chiral, anti-chiral, and hierarchical honeycombs: A simple energy-based approach. Theoretical and Applied Mechanics Letters, 6: 81-96. doi: 10.1016/j.taml.2016.02.004 [234] Mueller T, Hernandez A, Wang C. 2020. Machine learning for interatomic potential models. The Journal of Chemical Physics, 152: 050902. doi: 10.1063/1.5126336 [235] Mukhopadhyay T, Adhikari S. 2017a. Effective in-plane elastic moduli of quasi-random spatially irregular hexagonal lattices. International Journal of Engineering Science, 119: 142-179. doi: 10.1016/j.ijengsci.2017.06.004 [236] Mukhopadhyay T, Adhikari S. 2017b. Stochastic mechanics of metamaterials. Composite Structures, 162: 85-97. doi: 10.1016/j.compstruct.2016.11.080 [237] Mukhopadhyay T, Adhikari S. 2016a. Effective in-plane elastic properties of auxetic honeycombs with spatial irregularity. Mechanics of Materials, 95: 204-222. doi: 10.1016/j.mechmat.2016.01.009 [238] Mukhopadhyay T, Adhikari S. 2016b. Free-vibration analysis of sandwich panels with randomly irregular honeycomb core. Journal of Engineering Mechanics, 142: 06016008. [239] Mukhopadhyay T, Adhikari S. 2016c. Equivalent in-plane elastic properties of irregular honeycombs: An analytical approach. International Journal of Solids and Structures, 91: 169-184. doi: 10.1016/j.ijsolstr.2015.12.006 [240] Mukhopadhyay T, Adhikari S, Alu A. 2019a. Probing the frequency-dependent elastic moduli of lattice materials. Acta Materialia, 165: 654-665. doi: 10.1016/j.actamat.2018.11.012 [241] Mukhopadhyay T, Adhikari S, Alu A. 2019b. Theoretical limits for negative elastic moduli in subacoustic lattice materials. Phys. Rev. B, 99: 094108. doi: 10.1103/PhysRevB.99.094108 [242] Mukhopadhyay T, Adhikari S, Batou A. 2019c. Frequency domain homogenization for the viscoelastic properties of spatially correlated quasi-periodic lattices. International Journal of Mechanical Sciences, 150: 784-806. doi: 10.1016/j.ijmecsci.2017.09.004 [243] Mukhopadhyay T, Kundu D. 2022. Mixed-mode multidirectional Poisson’s ratio modulation in auxetic 3D lattice metamaterials. Advanced Engineering Materials, 24: 2101183. doi: 10.1002/adem.202101183 [244] Mukhopadhyay T, Ma J, Feng H, Hou D, Gattas J M, Chen Y, You Z. 2020a. Programmable stiffness and shape modulation in origami materials: Emergence of a distant actuation feature. Applied Materials Today, 19: 100537. doi: 10.1016/j.apmt.2019.100537 [245] Mukhopadhyay T, Mahata A, Adhikari S, Zaeem M A. 2018. Probing the shear modulus of two-dimensional multiplanar nanostructures and heterostructures. Nanoscale, 10: 5280-5294. doi: 10.1039/C7NR07261A [246] Mukhopadhyay T, Mahata A, Adhikari S, Zaeem M A. 2017a. Effective elastic properties of two dimensional multiplanar hexagonal nanostructures. 2D Mater., 4: 025006. doi: 10.1088/2053-1583/aa551c [247] Mukhopadhyay T, Mahata A, Adhikari S, Zaeem M A. 2017b. Effective mechanical properties of multilayer nano-heterostructures. Sci. Rep., 7: 15818. doi: 10.1038/s41598-017-15664-3 [248] Mukhopadhyay T, Mahata A, Naskar S, Adhikari S. 2020b. Probing the effective Young’s modulus of ‘magic angle’ inspired multi-functional twisted nano-heterostructures. Advanced Theory and Simulations, 3: 2000129. doi: 10.1002/adts.202000129 [249] Mukhopadhyay T, Naskar S, Adhikari S. 2020c. Anisotropy tailoring in geometrically isotropic multi-material lattices. Extreme Mechanics Letters, 40: 100934. doi: 10.1016/j.eml.2020.100934 [250] Mukhopadhyay T, Naskar S, Chakraborty S, Karsh P K, Choudhury R, Dey S. 2021. Stochastic oblique impact on composite laminates: A concise review and characterization of the essence of hybrid machine learning algorithms. Arch Computat Methods Eng, 28: 1731-1760. doi: 10.1007/s11831-020-09438-w [251] Münchinger A, Hsu L Y, Fürniß F, Blasco E, Wegener M. 2022. 3D optomechanical metamaterials. Materials Today, 59: 9-17. doi: 10.1016/j.mattod.2022.08.020 [252] Nabian M A, Meidani H. 2018. Deep learning for accelerated seismic reliability analysis of transportation networks. Computer-Aided Civil and Infrastructure Engineering, 33: 443-458. doi: 10.1111/mice.12359 [253] Narang Y S, Vlassak J J, Howe R D. 2018. Mechanically versatile soft machines through Laminar Jamming. Advanced Functional Materials, 28: 1707136. doi: 10.1002/adfm.201707136 [254] Nash L M, Kleckner D, Read A, Vitelli V, Turner A M, Irvine W T M. 2015. Topological mechanics of gyroscopic metamaterials. Proceedings of the National Academy of Sciences, 112: 14495-14500. doi: 10.1073/pnas.1507413112 [255] Neelakantan S, Bosbach W, Woodhouse J, Markaki A E. 2014. Characterization and deformation response of orthotropic fibre networks with auxetic out-of-plane behaviour. Acta Materialia, 66: 326-339. doi: 10.1016/j.actamat.2013.11.020 [256] Ngo T D, Kashani A, Imbalzano G, Nguyen K T Q, Hui D. 2018. Additive manufacturing (3D printing): A review of materials, methods, applications and challenges. Composites Part B: Engineering, 143: 172-196. doi: 10.1016/j.compositesb.2018.02.012 [257] Nguyen C, Zhuang X, Chamoin L, Zhao X, Nguyen-Xuan H, Rabczuk T. 2020. Three-dimensional topology optimization of auxetic metamaterial using isogeometric analysis and model order reduction. Computer Methods in Applied Mechanics and Engineering, 371: 113306. doi: 10.1016/j.cma.2020.113306 [258] Nick Z H, Tabor C E, Harne R L. 2020. Liquid metal microchannels as digital sensors in mechanical metamaterials. Extreme Mechanics Letters, 40: 100871. doi: 10.1016/j.eml.2020.100871 [259] Nicolaou Z G, Motter A E. 2012. Mechanical metamaterials with negative compressibility transitions. Nature Mater., 11: 608-613. doi: 10.1038/nmat3331 [260] Ning X, Wang X, Zhang Y, Yu X, Choi D, Zheng N, Kim D S, Huang Y, Zhang Y H, Rogers, J A. 2018. Assembly of advanced materials into 3d functional structures by methods inspired by origami and kirigami: A review. Advanced Materials Interfaces, 5: 1800284. doi: 10.1002/admi.201800284 [261] Nouh M, Aldraihem O, Baz A. 2014. Vibration characteristics of metamaterial beams with periodic local resonances. Journal of Vibration and Acoustics, 136 : 061012. [262] Novelino L S, Ze Q, Wu S, Paulino G H, Zhao R. 2020. Untethered control of functional origami microrobots with distributed actuation. Proceedings of the National Academy of Sciences, 117: 24096-24101. doi: 10.1073/pnas.2013292117 [263] O’Brien K, Suchowski H, Rho J, Salandrino A, Kante B, Yin X, Zhang X. 2015. Predicting nonlinear properties of metamaterials from the linear response. Nat. Mater., 14: 379-383. doi: 10.1038/nmat4214 [264] O’Connor H J, Dickson A N, Dowling D P. 2018. Evaluation of the mechanical performance of polymer parts fabricated using a production scale multi jet fusion printing process. Additive Manufacturing, 22: 381-387. doi: 10.1016/j.addma.2018.05.035 [265] Oh J H, Seung H M, Kim Y Y. 2016. Adjoining of negative stiffness and negative density bands in an elastic metamaterial. Applied Physics Letters, 108: 093501. doi: 10.1063/1.4943095 [266] Ongaro F. 2018. Estimation of the effective properties of two-dimensional cellular materials: A review. Theoretical and Applied Mechanics Letters, 8: 209-230. doi: 10.1016/j.taml.2018.04.010 [267] Overvelde J T B, de Jong T A, Shevchenko Y, Becerra S A, Whitesides G M, Weaver J C, Hoberman C, Bertoldi K. 2016. A three-dimensional actuated origami-inspired transformable metamaterial with multiple degrees of freedom. Nat. Commun., 7: 1-8. [268] Pahlavani H, Amani M, Saldívar M C, Zhou J, Mirzaali M J, Zadpoor A A. 2022. Deep learning for the rare-event rational design of 3D printed multi-material mechanical metamaterials. Commun Mater., 3: 1-11. doi: 10.1038/s43246-021-00223-1 [269] Palermo A, Marzani A. 2016. Extended bloch mode synthesis: Ultrafast method for the computation of complex band structures in phononic media. International Journal of Solids and Structures, 100: 29-40. [270] Palleau E, Morales D, Dickey M D, Velev O D. 2013. Reversible patterning and actuation of hydrogels by electrically assisted ionoprinting. Nat. Commun., 4: 2257. doi: 10.1038/ncomms3257 [271] Pan Q, Chen S, Chen F, Zhu X. 2020. Programmable soft bending actuators with auxetic metamaterials. Sci. China Technol. Sci., 63: 2518-2526. doi: 10.1007/s11431-020-1741-2 [272] Parthasarathy J, Starly B, Raman S, Christensen A. 2010. Mechanical evaluation of porous titanium (Ti6Al4V) structures with electron beam melting (EBM). Journal of the Mechanical Behavior of Biomedical Materials, 3: 249-259. doi: 10.1016/j.jmbbm.2009.10.006 [273] Paulose J, Chen B G, Vitelli V. 2015. Topological modes bound to dislocations in mechanical metamaterials. Nature Phys, 11: 153-156. doi: 10.1038/nphys3185 [274] Pendry J B. 2000. Negative refraction makes a perfect lens. Phys. Rev. Lett., 85: 3966-3969. doi: 10.1103/PhysRevLett.85.3966 [275] Peng X, Li Y, Zhang Q, Shang C, Bai Q W, Wang H. 2016. Tough hydrogels with programmable and complex shape deformations by ion dip-dyeing and transfer printing. Advanced Functional Materials, 26: 4491-4500. doi: 10.1002/adfm.201601389 [276] Poddubny A, Iorsh I, Belov P, Kivshar Y. 2013. Hyperbolic metamaterials. Nature Photon, 7: 948-957. doi: 10.1038/nphoton.2013.243 [277] Podolskiy V A, Kuhta N A, Milton G W. 2005. Optimizing the superlens: Manipulating geometry to enhance the resolution. Applied Physics Letters, 87: 231113. doi: 10.1063/1.2139620 [278] Prajwal P, Ghuku S, Mukhopadhyay T. 2022. Large-deformation mechanics of anti-curvature lattice materials for mode-dependent enhancement of non-linear shear modulus. Mechanics of Materials, 171: 104337. doi: 10.1016/j.mechmat.2022.104337 [279] Prall D, Lakes R S. 1997. Properties of a chiral honeycomb with a Poisson’s ratio of −1. International Journal of Mechanical Sciences, 39: 305-314. doi: 10.1016/S0020-7403(96)00025-2 [280] Pratapa P P, Suryanarayana P, Paulino G H. 2018. Bloch wave framework for structures with nonlocal interactions: Application to the design of origami acoustic metamaterials. Journal of the Mechanics and Physics of Solids, 118: 115-132. doi: 10.1016/j.jmps.2018.05.012 [281] Qi S, Oudich M, Li Y, Assouar, B. 2016. Acoustic energy harvesting based on a planar acoustic metamaterial. Applied Physics Letters, 108: 263501. doi: 10.1063/1.4954987 [282] Rafi H K, Karthik N V, Gong H, Starr T L, Stucker B E. 2013. Microstructures and mechanical properties of Ti6Al4V parts fabricated by selective laser melting and electron beam melting. J. of Materi Eng and Perform, 22: 3872-3883. doi: 10.1007/s11665-013-0658-0 [283] Rafsanjani A, Bertoldi K. 2017. Buckling-induced kirigami. Phys. Rev. Lett., 118: 084301. doi: 10.1103/PhysRevLett.118.084301 [284] Rafsanjani A, Jin L, Deng B, Bertoldi K. 2019. Propagation of pop ups in kirigami shells. Proceedings of the National Academy of Sciences, 116: 8200-8205. doi: 10.1073/pnas.1817763116 [285] Raghunath G, Flatau A B. 2015. Study of magnetic domain evolution in an auxetic plane of Galfenol using kerr microscopy. Journal of Applied Physics, 117: 17E704. doi: 10.1063/1.4913727 [286] Reid D R, Pashine N, Bowen A S, Nagel S R, Pablo J J de. 2019. Ideal isotropic auxetic networks from random networks. Soft Matter, 15: 8084-8091. doi: 10.1039/C9SM01241A [287] Ren Z, Hu W, Dong X, Sitti M. 2019. Multi-functional soft-bodied jellyfish-like swimming. Nat. Commun., 10: 2703. doi: 10.1038/s41467-019-10549-7 [288] Roach D, Hamel C, Dunn C, Johnson M, Kuang X, Qi H. 2019. The m4 3d printer: A multi-material multi-method additive manufacturing platform for future 3d printed structures. Additive Manufacturing, 29: 100819. doi: 10.1016/j.addma.2019.100819 [289] Robertson I D, Yourdkhani M, Centellas P J, Aw J E, Ivanoff D G, Goli E, Lloyd E M, Dean L M, Sottos N R, Geubelle P H, Moore J S, White S R. 2018. Rapid energy-efficient manufacturing of polymers and composites via frontal polymerization. Nature, 557: 223-227. doi: 10.1038/s41586-018-0054-x [290] Rogers J, Huang Y, Schmidt O G, Gracias D H. 2016. Origami mems and nems. MRS Bulletin, 41: 123-129. doi: 10.1557/mrs.2016.2 [291] Rothemund P W K. 2006. Folding DNA to create nanoscale shapes and patterns. Nature, 440: 297-302. doi: 10.1038/nature04586 [292] Rozvany G I N. 2009. A critical review of established methods of structural topology optimization. Struct Multidisc Optim, 37: 217-237. doi: 10.1007/s00158-007-0217-0 [293] Scarpa F, Bullough W A, Lumley P. 2004. Trends in acoustic properties of iron particle seeded auxetic polyurethane foam. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 218: 241-244. doi: 10.1243/095440604322887099 [294] Scarpa F, Panayiotou P, Tomlinson G. 2000. Numerical and experimental uniaxial loading on in-plane auxetic honeycombs. The Journal of Strain Analysis for Engineering Design, 35: 383-388. doi: 10.1243/0309324001514152 [295] Scarpa F, Smith F C. 2004. Passive and mr fluid-coated auxetic pu foam–mechanical, acoustic, and electromagnetic properties. Journal of Intelligent Material Systems and Structures, 15: 973-979. doi: 10.1177/1045389X04046610 [296] Schaedler T A, Carter W B. 2016. Architected cellular materials. Annual Review of Materials Research, 46: 187-210. doi: 10.1146/annurev-matsci-070115-031624 [297] Schaedler T A, Jacobsen A J, Torrents A, Sorensen A E, Lian J, Greer J R, Valdevit L, Carter W B. 2011. Ultralight metallic microlattices. Science, 334: 962-965. doi: 10.1126/science.1211649 [298] Schaeffer Marshall, Ruzzene M. 2015. Homogenization of 1d and 2d magnetoelastic lattices. EPJ Applied Metamaterials, 2: 13. doi: 10.1051/epjam/2015013 [299] Schaeffer M, Ruzzene M. 2015. Wave propagation in multistable magneto-elastic lattices. International Journal of Solids and Structures, 56: 78-95. [300] Schenk M, Guest S D. 2013. Geometry of miura-folded metamaterials. Proceedings of the National Academy of Sciences, 110: 3276-3281. doi: 10.1073/pnas.1217998110 [301] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R. 2006. Metamaterial electromagnetic cloak at microwave frequencies. Science, 314: 977-980. doi: 10.1126/science.1133628 [302] Serbin J, Ovsianikov A, Chichkov B. 2004. Fabrication of woodpile structures by two-photon polymerization and investigation of their optical properties. Opt. Express, 12: 5221-5228. doi: 10.1364/OPEX.12.005221 [303] Shadrivov I V. 2010. Nonlinear metamaterials. Nonlinearities in Periodic Structures and Metamaterials, Springer, pp: 241-257. [304] Shalaev V M. 2007. Optical negative-index metamaterials. Nat. Photoics, 1: 41-48. doi: 10.1038/nphoton.2006.49 [305] Shalaev V M, Cai W, Chettiar U K, Yuan H K, Sarychev A K, Drachev V P, Kildishev A V. 2005. Negative index of refraction in optical metamaterials. Opt. Lett, 30: 3356-3358. doi: 10.1364/OL.30.003356 [306] Shan S, Kang S H, Raney J R, Wang P, Fang L, Candido F, Lewis J A, Bertoldi K. 2015. Multistable architected materials for trapping elastic strain energy. Advanced Materials, 27: 4296-4301. doi: 10.1002/adma.201501708 [307] Sharma A, Mukhopadhyay T, Rangappa S M, Siengchin S, Kushvaha V. 2022. Advances in computational intelligence of polymer composite materials: Machine learning assisted modeling, analysis and design. Arch Computat Methods Eng, 29: 3341-3385. doi: 10.1007/s11831-021-09700-9 [308] Shelby R A, Smith D R, Schultz S. 2001. Experimental verification of a negative index of refraction. Science, 292: 77-79. doi: 10.1126/science.1058847 [309] Silva M J, Hayes W C, Gibson L J. 1995. The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids. International Journal of Mechanical Sciences, 37: 1161-1177. doi: 10.1016/0020-7403(94)00018-F [310] Silverberg J L, Evans A A, McLeod L, Hayward R C, Hull T, Santangelo C D, Cohen I. 2014. Using origami design principles to fold reprogrammable mechanical metamaterials. Science, 345: 647-650. doi: 10.1126/science.1252876 [311] Silverberg J L, Na J H, Evans A A, Liu B, Hull T C, Santangelo C D, Lang R J, Hayward R C, Cohen I. 2015. Origami structures with a critical transition to bistability arising from hidden degrees of freedom. Nat. Mater., 14: 389-393. doi: 10.1038/nmat4232 [312] Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2022a. Extreme on-demand contactless modulation of elastic properties in magnetostrictive lattices. Smart Mater. Struct., 31: 125005. doi: 10.1088/1361-665X/ac9cac [313] Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2022b. Active multi-physical modulation of Poisson’s ratios in composite piezoelectric lattices: On-demand sign reversal. Composite Structures, 280: 114857. doi: 10.1016/j.compstruct.2021.114857 [314] Singh A, Mukhopadhyay T, Adhikari S, Bhattacharya B. 2021. Voltage-dependent modulation of elastic moduli in lattice metamaterials: Emergence of a programmable state-transition capability. International Journal of Solids and Structures, 208: 31-48. [315] Singh K, Tipton C R, Han E, Mullin T. 2013. Magneto-elastic buckling of an Euler beam. Proc. R. Soc. A., 469: 20130111. doi: 10.1098/rspa.2013.0111 [316] Sinha A, Mukhopadhyay T. 2022c. Kirigami-inspired metamaterials for programming constitutive laws: Mixed-mode multidirectional auxeticity and contact-induced stiffness modulation. iScience, 25: 105656. doi: 10.1016/j.isci.2022.105656 [317] Sinha P, Mukhopadhyay T. 2023. On-demand contactless programming of nonlinear elastic moduli in hard magnetic soft beam based broadband active lattice materials. Smart Mater. Struct., 32: 055021. doi: 10.1088/1361-665X/acc43b [318] Sinha P, Mukhopadhyay T. 2022. Effective elastic properties of lattice materials with intrinsic stresses. Thin-Walled Structures, 173: 108950. doi: 10.1016/j.tws.2022.108950 [319] Sinha P, Mukhopadhyay T. 2023b. Elastostatics of inflatable lattices: realization of extreme specific stiffness along with multi-functionality in active modulation and deployability. In press. [320] Sinha P, Kundu D, Mukhopadhyay T. 2023a. Effective in-plane and out-of-plane elastic properties of 3d lattice materials with intrinsic stresses: an analytical approach. In press. [321] Sinha P, Walker M G, Mukhopadhyay T. 2023b. Non-invariant elastic moduli of bi-level architected lattice materials through programmed domain discontinuity. Mechanics of Materials, 184: 104691. doi: 10.1016/j.mechmat.2023.104691 [322] Slesarenko V. 2020. Planar mechanical metamaterials with embedded permanent magnets. Materials, 13: 1313. doi: 10.3390/ma13061313 [323] Smith D R, Pendry J B, Wiltshire M C K. 2004. Metamaterials and negative refractive index. Science, 305: 788-792. doi: 10.1126/science.1096796 [324] Song J, Gao L, Cao K, Zhang H, Xu S, Jiang C, Surjadi J U, Xu Y, Lu Y. 2018. Metal-coated hybrid meso-lattice composites and their mechanical characterizations. Composite Structures, 203: 750-763. doi: 10.1016/j.compstruct.2018.07.074 [325] Song Z, Ma T, Tang R, Cheng Q, Wang X, Krishnaraju D, Panat R, Chan C K, Yu H, Jiang H. 2014. Origami lithium-ion batteries. Nat. Commun., 5: 3140. doi: 10.1038/ncomms4140 [326] Soukoulis C M, Wegener M. 2010. Optical metamaterials—More bulky and less lossy. Science, 330: 1633-1634. doi: 10.1126/science.1198858 [327] Spadoni A, Ruzzene M. 2007. Numerical and experimental analysis of the static compliance of chiral truss-core airfoils. Journal Of Mechanics Of Materials And Structures, 2: 965-981. doi: 10.2140/jomms.2007.2.965 [328] Sugino C, Leadenham S, Ruzzene M, Erturk A. 2016. On the mechanism of bandgap formation in locally resonant finite elastic metamaterials. Journal of Applied Physics, 120: 134501. doi: 10.1063/1.4963648 [329] Sundararaghavan V, Zabaras N. 2005. Classification and reconstruction of three-dimensional microstructures using support vector machines. Computational Materials Science, 32: 223-239. doi: 10.1016/j.commatsci.2004.07.004 [330] Surjadi J U, Gao L, Du H, Li X, Xiong X, Fang N X, Lu Y. 2019. Mechanical metamaterials and their engineering applications. Advanced Engineering Materials, 21: 1800864. doi: 10.1002/adem.201800864 [331] Sussman D M, Cho Y, Castle T, Gong X, Jung E, Yang S, Kamien R D. 2015. Algorithmic lattice kirigami: A route to pluripotent materials. Proceedings of the National Academy of Sciences, 112: 7449-7453. doi: 10.1073/pnas.1506048112 [332] Tan X, Chen S, Wang B, Tang J, Wang L, Zhu S, Yao K, Xu P. 2020. Real-time tunable negative stiffness mechanical metamaterial. Extreme Mechanics Letters, 41: 100990. doi: 10.1016/j.eml.2020.100990 [333] Tan X, Wang B, Zhu S, Chen S, Yao K, Xu P, Wu L, Sun Y. 2019. Novel multidirectional negative stiffness mechanical metamaterials. Smart Materials and Structures, 29 : 015037. [334] Tang Y, Li Y, Hong Y, Yang S, Yin J. 2019. Programmable active kirigami metasheets with more freedom of actuation. Proceedings of the National Academy of Sciences, 116: 26407-26413. doi: 10.1073/pnas.1906435116 [335] Tao R, Ji L, Li Y, Wan Z, Hu W, Wu W, Liao B, Ma L, Fang D. 2020. 4D printed origami metamaterials with tunable compression twist behavior and stress-strain curves. Composites Part B: Engineering, 201: 108344. doi: 10.1016/j.compositesb.2020.108344 [336] Tee Y L, Peng C, Pille P, Leary M, Tran P. 2020. Polyjet 3d printing of composite materials: Experimental and modelling approach. JOM, 72: 1105-1117. doi: 10.1007/s11837-020-04014-w [337] Thijs L, Verhaeghe F, Craeghs T, Humbeeck J V, Kruth J P. 2010. A study of the microstructural evolution during selective laser melting of Ti–6Al–4V. Acta Materialia, 58: 3303-3312. doi: 10.1016/j.actamat.2010.02.004 [338] Thomson B K, William. 2010. Baltimore lectures on molecular dynamics and the wave theory of light, cambridge library collection-physical sciences. Cambridge University Press, Cambridge. [339] Tian Y, Shen Y. 2020. Selective guided wave mode transmission enabled by elastic metamaterials. Journal of Sound and Vibration, 485: 115566. doi: 10.1016/j.jsv.2020.115566 [340] Tipton C R, Han E, Mullin T. 2012. Magneto-elastic buckling of a soft cellular solid. Soft Matter, 8: 6880-6883. doi: 10.1039/c2sm25965f [341] Tiwari P, Naskar S, Mukhopadhyay T. 2023. Programmed out-of-plane curvature to enhance multimodal stiffness of bending-dominated composite lattices. AIAA Journal, 61: 1820-1838. doi: 10.2514/1.J062573 [342] Vaishali Mukhopadhyay T, Naskar S, Dey S. 2023. On machine learning assisted data-driven bridging of fsdt and hozt for high-fidelity uncertainty quantification of laminated composite and sandwich plates. Composite Structures, 304: 116276. doi: 10.1016/j.compstruct.2022.116276 [343] Valentine J, Zhang S, Zentgraf T, Ulin-Avila E, Genov D A, Bartal G, Zhang X. 2008. Three-dimensional optical metamaterial with a negative refractive index. Nature, 455: 376-379. doi: 10.1038/nature07247 [344] Valipour A, Kargozarfard M H, Rakhshi M, Yaghootian A, Sedighi H M. 2022. Metamaterials and their applications: An overview. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 236: 2171-2210. doi: 10.1177/1464420721995858 [345] van Manen T, Janbaz S, Ganjian M, Zadpoor A A. 2020. Kirigami-enabled self-folding origami. Materials Today, 32: 59-67. doi: 10.1016/j.mattod.2019.08.001 [346] Vangelatos Z, Gu G X, Grigoropoulos C P. 2019. Architected metamaterials with tailored 3d buckling mechanisms at the microscale. Extreme Mechanics Letters, 33: 100580. doi: 10.1016/j.eml.2019.100580 [347] Vyatskikh A, Delalande S, Kudo A, Zhang X, Portela C M, Greer J R. 2018. Additive manufacturing of 3D nano-architected metals. Nat. Commun., 9: 593. doi: 10.1038/s41467-018-03071-9 [348] Waheed U, Myant C W, Dobson S N. 2020. Boolean and/or mechanical logic using multi-plane mechanical metamaterials. Extreme Mechanics Letters, 40: 100865. doi: 10.1016/j.eml.2020.100865 [349] Waitukaitis S, Menaut R, Chen B G, van Hecke M. 2015. Origami multistability: From single vertices to metasheets. Phys. Rev. Lett., 114: 055503. doi: 10.1103/PhysRevLett.114.055503 [350] Wang A J, McDowell D L. 2003. Effects of defects on in-plane properties of periodic metal honeycombs. International Journal of Mechanical Sciences, 45: 1799-1813. doi: 10.1016/j.ijmecsci.2003.12.007 [351] Wang C, Tan X P, Tor S B, Lim C S. 2020. Machine learning in additive manufacturing: State-of-the-art and perspectives. Additive Manufacturing, 36: 101538. doi: 10.1016/j.addma.2020.101538 [352] Wang H, Zhao D, Jin Y, Wang M, Mukhopadhyay T, You Z. 2020. Modulation of multi-directional auxeticity in hybrid origami metamaterials. Applied Materials Today, 20: 100715. doi: 10.1016/j.apmt.2020.100715 [353] Wang L C, Song W L, Zhang Y J, Qu M J, Zhao Z, Chen M, Yang Y, Chen H, Fang D. 2020. Active reconfigurable tristable square-twist origami. Advanced Functional Materials, 30: 1909087. doi: 10.1002/adfm.201909087 [354] Wang P, Shim J, Bertoldi K. 2013. Effects of geometric and material nonlinearities on tunable band gaps and low-frequency directionality of phononic crystals. Phys. Rev. B, 88: 014304. doi: 10.1103/PhysRevB.88.014304 [355] Wang S, Shen Z A, Shen Z Y, Dong Y, Li Y, Cao Y, Zhang Y, Guo S, Shuai J, Yang Y, Lin C, Chen X, Zhang X, Huang Q. 2021. Machine-learning micropattern manufacturing. Nano Today, 38: 101152. doi: 10.1016/j.nantod.2021.101152 [356] Wang W, Lu H, Liu Y, Leng J. 2014. Sodium dodecyl sulfate/epoxy composite: Water-induced shape memory effect and its mechanism. J. Mater. Chem. A, 2: 5441-5449. doi: 10.1039/c3ta15204a [357] Wang X Q, Chan K H, Cheng Y, Ding T, Li T, Achavananthadith S, Ahmet S, Ho J S, Ho G W. 2020. Somatosensory, light-driven, thin-film robots capable of integrated perception and motility. Advanced Materials, 32: 2000351. doi: 10.1002/adma.202000351 [358] Wang Z, Li K, He Q, Cai S. 2019. A light-powered ultralight tensegrity robot with high deformability and load capacity. Advanced Materials, 31: 1806849. doi: 10.1002/adma.201806849 [359] Wang Z J, Zhu C N, Hong W, Wu Z L, Zheng Q. 2017. Cooperative deformations of periodically patterned hydrogels. Science Advances, 3: e1700348. doi: 10.1126/sciadv.1700348 [360] Ward L, Agrawal A, Choudhary A, Wolverton C. 2016. A general-purpose machine learning framework for predicting properties of inorganic materials. npj Comput Mater., 2: 1-7. doi: 10.1038/s41524-016-0001-z [361] Wei Y L, Yang Q S, Ma L H, Tao R, Shang J J. 2020. Design and analysis of 2d/3d negative hydration expansion metamaterial driven by hydrogel. Materials & Design, 196: 109084. [362] Wilt J K, Yang C, Gu G X. 2020. Accelerating auxetic metamaterial design with deep learning. Advanced Engineering Materials, 22: 2070018. doi: 10.1002/adem.202070018 [363] Wu L, Liu L, Wang Y, Zhai Z, Zhuang H, Krishnaraju D, Wang Q, Jiang H. 2020. A machine learning-based method to design modular metamaterials. Extreme Mechanics Letters, 36: 100657. doi: 10.1016/j.eml.2020.100657 [364] Wu R, Roberts P C E, Lyu S, Zheng F, Soutis C, Diver C, Zhou D, Li L, Deng Z. 2021. Lightweight self-forming super-elastic mechanical metamaterials with adaptive stiffness. Advanced Functional Materials, 31: 2008252. doi: 10.1002/adfm.202008252 [365] Wu W, Tao Y, Xia Y, Chen J, Lei H, Sun L, Fang D. 2017. Mechanical properties of hierarchical anti-tetrachiral metastructures. Extreme Mechanics Letters, 16: 18-32. doi: 10.1016/j.eml.2017.08.004 [366] Wu X, Jin Y, Khelif A, Zhuang X, Rabczuk T, Djafari-Rouhani B. 2022. Topological surface wave metamaterials for robust vibration attenuation and energy harvesting. Mechanics of Advanced Materials and Structures, 29: 4759-4767. doi: 10.1080/15376494.2021.1937758 [367] Wyart M, Liang H, Kabla A, Mahadevan L. 2008. Elasticity of floppy and stiff random networks. Phys. Rev. Lett., 101: 215501. doi: 10.1103/PhysRevLett.101.215501 [368] Xia X, Afshar A, Yang H, Portela C M, Kochmann D M, Di Leo C V, Greer J R. 2019. Electrochemically reconfigurable architected materials. Nature, 573: 205-213. doi: 10.1038/s41586-019-1538-z [369] Xia X, Spadaccini C M, Greer J R. 2022. Responsive materials architected in space and time. Nat. Rev. Mater., 7: 683-701. doi: 10.1038/s41578-022-00450-z [370] Xiang X M, Lu G, You Z. 2020. Energy absorption of origami inspired structures and materials. Thin-Walled Structures, 157: 107130. doi: 10.1016/j.tws.2020.107130 [371] Xin X, Liu L, Liu Y, Leng J. 2020. Origami-inspired self-deployment 4d printed honeycomb sandwich structure with large shape transformation. Smart Mater. Struct., 29: 065015. doi: 10.1088/1361-665X/ab85a4 [372] Xu C, Quinn B, L’Espérance G, Lebel L, Daniel T. 2019. Multi-material direct ink writing (DIW) for complex 3d metallic structures with removable supports. ACS Applied Materials & Interfaces, 11: 8499-8506. [373] Xu L, Wang X, Kim Y, Shyu T C, Lyu J, Kotov N A. 2016. Kirigami nanocomposites as wide-angle diffraction gratings. Acs Nano, 10: 6156-6162. doi: 10.1021/acsnano.6b02096 [374] Yang C, Boorugu M, Dopp A, Ren J, Martin R, Han D, Choi W, Lee H. 2019. 4d printing reconfigurable, deployable and mechanically tunable metamaterials. Mater. Horiz., 6: 1244-1250. doi: 10.1039/C9MH00302A [375] Yang H, D’Ambrosio N, Liu P, Pasini D, Ma L. 2023. Shape memory mechanical metamaterials. Materials Today, 66: 36-49. doi: 10.1016/j.mattod.2023.04.003 [376] Yang H, Ma L. 2020. 1d to 3d multi-stable architected materials with zero Poisson’s ratio and controllable thermal expansion. Materials & Design, 188: 108430. [377] Yang H, Ma L. 2019. Multi-stable mechanical metamaterials by elastic buckling instability. J Mater Sci, 54: 3509-3526. doi: 10.1007/s10853-018-3065-y [378] Yang L, Harrysson O, Cormier D, West H, Gong H, Stucker B. 2015. Additive manufacturing of metal cellular structures: Design and fabrication. JOM, 67: 608-615. doi: 10.1007/s11837-015-1322-y [379] Yang Y, You Z. 2020. A Modular Origami-inspired Mechanical Metamaterial. arXiv:2012.09567 [380] Yao J, Sun R, Scarpa F, Remillat C, Gao Y, Su Y. 2021. Two-dimensional graded metamaterials with auxetic rectangular perforations. Composite Structures, 261: 113313. doi: 10.1016/j.compstruct.2020.113313 [381] Yu K, Du H, Xin A, Lee K H, Feng Z, Masri S F, Chen Y, Huang G, Wang Q. 2020. Healable, memorizable, and transformable lattice structures made of stiff polymers. NPG Asia Mater., 12: 1-16. doi: 10.1038/s41427-019-0187-x [382] Yu X, Zhou J, Liang H, Jiang Z, Wu L. 2018. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review. Progress in Materials Science, 94: 114-173. doi: 10.1016/j.pmatsci.2017.12.003 [383] Zadpoor A A. 2016. Mechanical meta-materials. Mater. Horiz., 3: 371-381. doi: 10.1039/C6MH00065G [384] Zeng C, Liu L, Bian W, Leng J, Liu Y. 2022. Temperature-dependent mechanical response of 4d printed composite lattice structures reinforced by continuous fiber. Composite Structures, 280: 114952. doi: 10.1016/j.compstruct.2021.114952 [385] Zeng H, Wani O M, Wasylczyk P, Priimagi A. 2018. Light-driven, caterpillar-inspired miniature inching robot. Macromolecular Rapid Communications, 39: 1700224. doi: 10.1002/marc.201700224 [386] Zhai Z, Wang Y, Jiang H. 2018. Origami-inspired, on-demand deployable and collapsible mechanical metamaterials with tunable stiffness. Proceedings of the National Academy of Sciences, 115: 2032-2037. doi: 10.1073/pnas.1720171115 [387] Zhai Z, Wang Y, Lin K, Wu L, Jiang H. 2020. In situ stiffness manipulation using elegant curved origami. Science Advances, 6: eabe2000. doi: 10.1126/sciadv.abe2000 [388] Zhai Z, Wu L, Jiang H. 2021. Mechanical metamaterials based on origami and kirigami. Applied Physics Reviews, 8: 041319. doi: 10.1063/5.0051088 [389] Zhang J, Ashby M F. 1992. The out-of-plane properties of honeycombs. International Journal of Mechanical Sciences, 34: 475-489. doi: 10.1016/0020-7403(92)90013-7 [390] Zhang J, Lu G, You Z. 2020. Large deformation and energy absorption of additively manufactured auxetic materials and structures: A review. Composites Part B: Engineering, 201: 108340. doi: 10.1016/j.compositesb.2020.108340 [391] Zhang K, Chermprayong P, Xiao F, Tzoumanikas D, Dams B, Kay S, Kocer B B, Burns A, Orr L, Choi C, Darekar D D, Li W, Hirschmann S, Soana V, Ngah S A, Sareh S, Choubey A, Margheri L, Pawar V M, Ball R J, Williams C, Shepherd P, Leutenegger S, Stuart-Smith R, Kovac M. 2022. Aerial additive manufacturing with multiple autonomous robots. Nature, 609: 709-717. doi: 10.1038/s41586-022-04988-4 [392] Zhang Q, Barri K, Jiao P, Lu W, Luo J, Meng W, Wang J, Hong L, Mueller J, Lin Wang Z, Alavi A H. 2023. Meta-mechanotronics for self-powered computation. Materials Today, 65: 78-89. doi: 10.1016/j.mattod.2023.03.026 [393] Zhang Q, Kuang X, Weng S, Zhao Z, Chen H, Fang D, Qi H J. 2020. Rapid volatilization induced mechanically robust shape-morphing structures toward 4d printing. ACS Appl. Mater. Interfaces, 12: 17979-17987. doi: 10.1021/acsami.0c02038 [394] Zhang Q, Wommer J, O’Rourke C, Teitelman J, Tang Y, Robison J, Lin G, Yin J. 2017. Origami and kirigami inspired self-folding for programming three-dimensional shape shifting of polymer sheets with light. Extreme Mechanics Letters, 11: 111-120. doi: 10.1016/j.eml.2016.08.004 [395] Zhang S, Yin L, Fang N. 2009. Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett., 102: 194301. doi: 10.1103/PhysRevLett.102.194301 [396] Zhang X, Liu Z. 2008. Superlenses to overcome the diffraction limit. Nature Mater., 7: 435-441. doi: 10.1038/nmat2141 [397] Zhang Y, Wang Q, Tichem M, van Keulen F. 2020. Design and characterization of multi-stable mechanical metastructures with level and tilted stable configurations. Extreme Mechanics Letters, 34: 100593. doi: 10.1016/j.eml.2019.100593 [398] Zhang Y, Yan Z, Nan K, Xiao D, Liu Y, Luan H, Fu H, Wang X, Yang Q, Wang J C, Ren W, Si H, Liu F, Yang L, Li H, Wang J T, Guo X, Luo H, Wang L, Huang Y, Rogers J A. 2015. A mechanically driven form of kirigami as a route to 3d mesostructures in micro/nanomembranes. Proceedings of the National Academy of Sciences, 112: 11757-11764. doi: 10.1073/pnas.1515602112 [399] Zhang Z, Dou J, He J, Xiao C, Shen L, Yang J, Wang Y, Zhou Z. 2017. Electrically/infrared actuated shape memory composites based on a bio-based polyester blend and graphene nanoplatelets and their excellent self-driven ability. J. Mater. Chem. C, 5: 4145-4158. doi: 10.1039/C7TC00828G [400] Zhang Z, Krushynska A O. 2022. Programmable shape morphing of rose mechanical metamaterials. APL Mater., 10: 080701. doi: 10.1063/5.0099323 [401] Zhang Z, Scarpa F, Bednarcyk B A, Chen Y. 2021. Harnessing fractal cuts to design robust lattice metamaterials for energy dissipation. Additive Manufacturing, 46: 102126. doi: 10.1016/j.addma.2021.102126 [402] Zhao Z, Wu J, Mu X, Chen H, Qi H J, Fang D. 2017. Origami by frontal photopolymerization. Science Advances, 3: e1602326. doi: 10.1126/sciadv.1602326 [403] Zheng X, Lee H, Weisgraber T H, Shusteff M, DeOtte J, Duoss E B, Kuntz J D, Biener M M, Ge Q, Jackson J A, Kucheyev S O, Fang N X, Spadaccini C M. 2014. Ultralight, ultrastiff mechanical metamaterials. Science, 344: 1373-1377. doi: 10.1126/science.1252291 [404] Zhong H, Song T, Li C, Das R, Gu J, Qian M. 2023. The gibson-ashby model for additively manufactured metal lattice materials: Its theoretical basis, limitations and new insights from remedies. Current Opinion in Solid State and Materials Science, 27: 101081. doi: 10.1016/j.cossms.2023.101081 [405] Zhou X H, Li T, Wang J, Chen F, Zhou D, Liu Q, Li B, Cheng J, Zhou X C, Zheng B. 2018. Mechanochemical regulated origami with tough hydrogels by ion transfer printing. ACS Appl. Mater. Interfaces, 10: 9077-9084. doi: 10.1021/acsami.8b01610 [406] Zhu H X, Hobdell J R, Windle A H. 2001. Effects of cell irregularity on the elastic properties of 2d voronoi honeycombs. Journal of the Mechanics and Physics of Solids, 49: 857-870. doi: 10.1016/S0022-5096(00)00046-6 [407] Zhu R, Huang G L, Huang H H, Sun C T. 2011. Experimental and numerical study of guided wave propagation in a thin metamaterial plate. Physics Letters A, 375: 2863-2867. doi: 10.1016/j.physleta.2011.06.006 [408] Zhu S, Wang B, Tan X, Hu J, Wang L, Zhou Z, Chen S. 2021. A novel bi-material negative stiffness metamaterial in sleeve-type via combining rigidity with softness. Composite Structures, 262: 113381. doi: 10.1016/j.compstruct.2020.113381 [409] Zhu Y, Birla M, Oldham K R, Filipov E T. 2020. Elastically and plastically foldable electrothermal micro-origami for controllable and rapid shape morphing. Advanced Functional Materials, 30: 2003741. doi: 10.1002/adfm.202003741