Design strategies for non-positive Poisson's ratio mechanical metamaterials and their cushioning and energy absorption characteristics

WANG Weijing; YANG Hang; ZHANG Weiming; MA Li

doi:10.6052/1000-0992-25-021

Article Contents

Article Navigation > Advances in Mechanics > 2025 > Accepted Manuscript

Wang W J, Yang H, Zhang W M, Ma L. Design strategies for non-positive Poisson's ratio mechanical metamaterials and their cushioning and energy absorption characteristics. Advances in Mechanics, in press doi: 10.6052/1000-0992-25-021

Citation:

Wang W J, Yang H, Zhang W M, Ma L. Design strategies for non-positive Poisson's ratio mechanical metamaterials and their cushioning and energy absorption characteristics. Advances in Mechanics, in press doi: 10.6052/1000-0992-25-021

Citation:

PDF( 15141 KB)

Design strategies for non-positive Poisson's ratio mechanical metamaterials and their cushioning and energy absorption characteristics

doi: 10.6052/1000-0992-25-021 cstr: 32046.14.1000-0992-25-021

1.
Center for Composite Materials, Harbin Institute of Technology, Harbin 150080, China
2.
Department of Mechanical Engineering, National University of Singapore, Queenstown 117575, Singapore

More Information

Corresponding author: mali@hit.edu.cn
Received Date: 2025-08-19
Accepted Date: 2025-10-23

Available Online: 2025-11-05

Abstract

Abstract

Non-positive Poisson's ratio mechanical metamaterials are a class of architected functional materials that exhibit negative or zero Poisson's ratio effect at the macroscopic scale through configuration design. Their distinctive capabilities in controlling transverse deformation, maintaining dimensional stability, and enhancing energy absorption confer significant potential for applications in aerospace, marine engineering, transportation, wearable protective equipment, and biomedicine. In recent years, continuous advancements in microstructural design, advanced material fabrication techniques, and multi-material integration methods have driven significant progress in non-positive Poisson's ratio mechanical metamaterials, particularly in configuration diversity, mechanical response tunability, and multifunctional integration. Guided by the dominant mechanisms that activate transverse deformation, this paper systematically surveys the typical design strategies of non-positive Poisson's ratio mechanical metamaterials. For negative Poisson's ratio architectures, the discussion is organized around re-entrant geometries, rotating systems (rotating rigid-body/truss and chiral/anti-chiral configurations), kirigami/origami schemes, elastic instability-induced mechanisms, and rigid-body linkages. Zero Poisson's ratio architectures are categorized into geometric paradigms, including rectangular/parallelogram-like, semi-re-entrant, positive-negative Poisson's ratio unit hybrids, and rigid-flexible composites. Focusing on performance requirements in cushioning and energy absorption, enhancement strategies include multi-plateau response designs, graded structural architectures, multi-material coupling, and the incorporation of smart materials. At the level of structural integration, technical pathways such as modular assembly, sandwich structure, and intrinsically three-dimensional architectures are reviewed. Finally, by synthesizing recent research progress on non-positive Poisson's ratio mechanical metamaterials in terms of design and fabrication, performance regulation, and system integration, the current core technical bottlenecks are identified, the key directions for breakthroughs are clarified, and future development pathways for multiscale manufacturing, multifield response integration, and engineering applications are proposed.
- mechanical metamaterials,
- non-positive Poisson's ratio,
- microstructural design,
- cushioning and energy absorption,
- multi-material integration

FullText(HTML)

References(174)

References

[1]	廖瑜, 石少卿, 夏菲, 等. 2025. 双梯形负泊松比蜂窝夹心结构抗爆力学性能研究. 工程力学, 42: 244-261 (Liao Y, Shi S Q, Xia F, et al. 2025. Study on anti-explosion mechanical properties of double trapezoidal negative Poisson's ratio auxetic sandwich honeycomb structure. Engineering Mechanics, 42: 244-261). Liao Y, Shi S Q, Xia F, et al. 2025. Study on anti-explosion mechanical properties of double trapezoidal negative Poisson's ratio auxetic sandwich honeycomb structure. Engineering Mechanics, 42: 244-261
[2]	王钦泽, 韩宾, 郑培远, 等. 2024. 负刚度扭转超结构力学性能研究. 应用数学和力学, 45: 1082-1095 (Wang Q Z, Han B, Zheng P Y, et al. 2024. Research on mechanical properties of negative stiffness torsion metastructures. Applied Mathematics and Mechanics, 45: 1082-1095). Wang Q Z, Han B, Zheng P Y, et al. 2024. Research on mechanical properties of negative stiffness torsion metastructures. Applied Mathematics and Mechanics, 45: 1082-1095
[3]	王玮婧, 张伟明, 郭孟甫, 等. 2024. 内凹-星型三维负泊松比结构设计及冲击吸能特性. 振动与冲击, 43: 75-83 (Wang W J, Zhang W M, Guo M F, et al. 2024. Design and impact energy absorption characteristics of concave-star three dimensional negative Poisson's ratio structures. Journal of Vibration and Shock, 43: 75-83). Wang W J, Zhang W M, Guo M F, et al. 2024. Design and impact energy absorption characteristics of concave-star three dimensional negative Poisson's ratio structures. Journal of Vibration and Shock, 43: 75-83
[4]	王信涛. 2018. 三维有序负泊松比结构的设计、制备与力学性能表征[博士学位论文]. 哈尔滨: 哈尔滨工业大学 ((Wang X T. 2018. The design, fabrication and mechanical characterization of three-dimensional periodic auxetic cellular structures[PhD Thesis]. Harbin: Harbin Institute of Technology). (Wang X T. 2018. The design, fabrication and mechanical characterization of three-dimensional periodic auxetic cellular structures[PhD Thesis]. Harbin: Harbin Institute of Technology
[5]	薛玉祥. 2021. 三维负泊松比星型结构冲击动力学研究[硕士学位论文]. 广州: 广州大学 ((Xue Y X. 2021. Study on the impact dynamics of three-dimensional negative Poisson's ratio star-shaped structure[Master Thesis]. Guangzhou: Guangzhou University). (Xue Y X. 2021. Study on the impact dynamics of three-dimensional negative Poisson's ratio star-shaped structure[Master Thesis]. Guangzhou: Guangzhou University
[6]	杨航. 2023. 可编程机械超材料的结构设计及力学行为研究[博士学位论文]. 哈尔滨: 哈尔滨工业大学 ((Yang H. 2023. Structure design and mechanical behavior of programmable mechanical metamaterials[PhD Thesis]. Harbin: Harbin Institute of Technology). (Yang H. 2023. Structure design and mechanical behavior of programmable mechanical metamaterials[PhD Thesis]. Harbin: Harbin Institute of Technology
[7]	于靖军, 谢岩, 裴旭. 2018. 负泊松比超材料研究进展. 机械工程学报, 54: 1-14 (Yu J J, Xie Y, Pei X. 2018. State-of-art of metamaterials with negative Poisson's ratio. Journal of Mechanical Engineering, 54: 1-14). Yu J J, Xie Y, Pei X. 2018. State-of-art of metamaterials with negative Poisson's ratio. Journal of Mechanical Engineering, 54: 1-14
[8]	余阳, 付涛. 2023. 低速冲击下负泊松比蝴蝶形蜂窝夹芯板的动力响应. 爆炸与冲击, 43: 84-95 (Yu Y, Fu T. 2023. Dynamic response of a sandwich panel cored by butterfly-shaped honeycombs with negative Poisson's ratio to low-velocity impact. Explosion and Shock Waves, 43: 84-95). Yu Y, Fu T. 2023. Dynamic response of a sandwich panel cored by butterfly-shaped honeycombs with negative Poisson's ratio to low-velocity impact. Explosion and Shock Waves, 43: 84-95
[9]	张宝庆, 蒋森. 2025. 旋转型负泊松比星形蜂窝结构能量吸收特性研究. 固体力学学报, 46: 129-148 (Zhang B Q, Jiang S. 2025. Study on energy absorption characteristics of rotating star-shaped honeycomb structure with negative Poisson's ratio. Chinese Journal of Solid Mechanics, 46: 129-148). Zhang B Q, Jiang S. 2025. Study on energy absorption characteristics of rotating star-shaped honeycomb structure with negative Poisson's ratio. Chinese Journal of Solid Mechanics, 46: 129-148
[10]	赵淳铮, 王昕, 李振, 等. 2024. 可调控热膨胀力学超材料设计制备与表征评测研究进展. 复合材料学报, 41: 4589-4605 (Zhao C Z, Wang X, Li Z, et al. 2024. Research progress in the design, manufacturing, characterization, and evaluation of tailorable thermal expansion mechanical metamaterials. Acta Materiae Compositae Sinica, 41: 4589-4605). Zhao C Z, Wang X, Li Z, et al. 2024. Research progress in the design, manufacturing, characterization, and evaluation of tailorable thermal expansion mechanical metamaterials. Acta Materiae Compositae Sinica, 41: 4589-4605
[11]	Airoldi A, Novak N, Sgobba F, et al. 2020. Foam-filled energy absorbers with auxetic behaviour for localized impacts. Materials Science and Engineering: A, 788: 139500 doi: 10.1016/j.msea.2020.139500
[12]	Alderson A, Alderson K L, Attard D, et al. 2010. Elastic constants of 3-, 4- and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading. Composites Science and Technology, 70: 1042-1048 doi: 10.1016/j.compscitech.2009.07.009
[13]	Alderson K L, Webber R S, Evans K E. 2000. Novel variations in the microstructure of auxetic ultra-high molecular weight polyethylene. Part 2: Mechanical properties. Polymer Engineering & Science, 40: 1906-1914
[14]	Andrade C, Ha C S, Lakes R S. 2018. Extreme Cosserat elastic cube structure with large magnitude of negative Poisson's ratio. Journal of Mechanics of Materials and Structures, 13: 93-101 doi: 10.2140/jomms.2018.13.93
[15]	Bauer J, Schroer A, Schwaiger R, et al. 2016. Approaching theoretical strength in glassy carbon nanolattices. Nature Materials, 15: 438-443 doi: 10.1038/nmat4561
[16]	Baughman R H, Stafstrom S, Cui C, et al. 1998. Materials with negative compressibilities in one or more dimensions. Science, 279: 1522-1524 doi: 10.1126/science.279.5356.1522
[17]	Bertoldi K, Reis P M, Willshaw S, et al. 2010. Negative Poisson's ratio behavior induced by an elastic instability. Advanced Materials, 22: 361-366 doi: 10.1002/adma.200901956
[18]	Chen C Q, Airoldi A, Caporale A M, et al. 2024. Impact response of composite energy absorbers based on foam-filled metallic and polymeric auxetic frames. Composite Structures, 331: 117916 doi: 10.1016/j.compstruct.2024.117916
[19]	Chen C Q, He Y L, Xu R, et al. 2024. Dynamic behaviors of sandwich panels with 3D-printed gradient auxetic cores subjected to blast load. International Journal of Impact Engineering, 188: 104943 doi: 10.1016/j.ijimpeng.2024.104943
[20]	Chen C Q, Jiang L, Wang H R, et al. 2024. Quasi-static and dynamic responses of gradient hexachiral auxetics: Experimental and numerical analysis. Materials Today Communications, 41: 110670 doi: 10.1016/j.mtcomm.2024.110670
[21]	Chen Y, Fu M H. 2018. Mechanical properties of a novel zero Poisson's ratio honeycomb. Advanced Engineering Materials, 20: 1700452 doi: 10.1002/adem.201700452
[22]	Cui H, Hensleigh R, Yao D, et al. 2019. Three-dimensional printing of piezoelectric materials with designed anisotropy and directional response. Nature Materials, 18: 234-241 doi: 10.1038/s41563-018-0268-1
[23]	De Jong M, Chen W, Angsten T, et al. 2015. Charting the complete elastic properties of inorganic crystalline compounds. Scientific Data, 2: 150009 doi: 10.1038/sdata.2015.9
[24]	Del Broccolo S, Laurenzi S, Scarpa F. 2017. AuxHex-A Kirigami inspired zero Poisson's ratio cellular structure. Composite Structures, 176: 433-441 doi: 10.1016/j.compstruct.2017.05.050
[25]	Ding L, Zhang D, Lin Y, et al. 2024. Improving energy absorption of rotating triangle auxetic metamaterials through multistage perforations. Materials Today Communications, 41: 110746 doi: 10.1016/j.mtcomm.2024.110746
[26]	Dong Y, Huang N. 2024. Multifield and higher-order analysis of sandwich smart curved beams made of graphene origami auxetic metamaterial. Mechanics of Advanced Materials and Structures, 31: 1-19 doi: 10.1080/15376494.2024.2302247
[27]	Donoghue J, Alderson K, Evans K. 2009. The fracture toughness of composite laminates with a negative Poisson's ratio. Physica Status Solidi B, 246: 2011-2017 doi: 10.1002/pssb.200982031
[28]	Esmaeili A, Karimi M, Heidari-Rarani M, et al. 2024. A new design of star auxetic metastructure with enhanced energy-absorption under various loading rates: Experimental and numerical study. Structures, 63: 106457 doi: 10.1016/j.istruc.2024.106457
[29]	Etemadi E, Bashtani M, Hu H. 2024. Novel auxetic metamaterials inspired from geometry patterns of an Iranian Mosque with improved energy absorption capability. Materials Today Communications, 41: 110470 doi: 10.1016/j.mtcomm.2024.110470
[30]	Evans K E. 1991. Auxetic polymers: A new range of materials. Endeavour, 15: 170-174 doi: 10.1016/0160-9327(91)90123-S
[31]	Evans K E, Alderson A. 2000. Auxetic materials: Functional materials and structures from lateral thinking! Advanced Materials, 12: 617-628
[32]	Fan J, Zhang L, Wei S, et al. 2021. A review of additive manufacturing of metamaterials and developing trends. Materials Today, 50: 303-328 doi: 10.1016/j.mattod.2021.04.019
[33]	Fu M H, Zheng B B, Li W H. 2017. A novel chiral three-dimensional material with negative Poisson's ratio and the equivalent elastic parameters. Composite Structures, 176: 442-448 doi: 10.1016/j.compstruct.2017.05.027
[34]	Fu M, Liu F, Hu L. 2018. A novel category of 3D chiral material with negative Poisson's ratio. Composites Science and Technology, 160: 111-118 doi: 10.1016/j.compscitech.2018.03.017
[35]	Fu T, Hu X, Yang C. 2023. Impact response analysis of stiffened sandwich functionally graded porous materials doubly-curved shell with re-entrant honeycomb auxetic core. Applied Mathematical Modelling, 124: 553-575 doi: 10.1016/j.apm.2023.08.024
[36]	Fu T, Wang X, Hu X, et al. 2024. Impact dynamic response of stiffened porous functionally graded materials sandwich doubly-curved shells with arc-type auxetic core. International Journal of Impact Engineering, 191: 105000 doi: 10.1016/j.ijimpeng.2024.105000
[37]	Gao Y, Wei X, Han X, et al. 2021. Novel 3D auxetic lattice structures developed based on the rotating rigid mechanism. International Journal of Solids and Structures, 233: 111232 doi: 10.1016/j.ijsolstr.2021.111232
[38]	Gibson L J, Ashby M F, Schajer G, et al. 1982. The mechanics of two-dimensional cellular materials. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 382: 25-42 doi: 10.1098/rspa.1982.0087
[39]	Gömöry F, Solovyov M, Šouc J, et al. 2012. Experimental realization of a magnetic cloak. Science, 335: 1466-1468 doi: 10.1126/science.1218316
[40]	Gong X, Huang J, Scarpa F, et al. 2015. Zero Poisson's ratio cellular structure for two-dimensional morphing applications. Composite Structures, 134: 384-392 doi: 10.1016/j.compstruct.2015.08.048
[41]	Greaves G N, Greer A L, Lakes R S, et al. 2011. Poisson's ratio and modern materials. Nature Materials, 10: 823-837 doi: 10.1038/nmat3134
[42]	Grima J N, Evans K E. 2000. Auxetic behavior from rotating squares. Journal of Materials Science Letters, 19: 1563-1565 doi: 10.1023/A:1006781224002
[43]	Grima J N, Gatt R, Farrugia P S. 2008. On the properties of auxetic meta-tetrachiral structures. Physica Status Solidi B, 245: 511-520 doi: 10.1002/pssb.200777704
[44]	Grima J N, Mizzi L, Azzopardi K M, et al. 2016. Auxetic perforated mechanical metamaterials with randomly oriented cuts. Advanced Materials, 28: 385-389 doi: 10.1002/adma.201503653
[45]	Grima J N, Oliveri L, Attard D, et al. 2010. Hexagonal honeycombs with zero Poisson's ratios and enhanced stiffness. Advanced Engineering Materials, 12: 855-862 doi: 10.1002/adem.201000140
[46]	Günaydın K, Rea C, Kazancı Z. 2022. Energy absorption enhancement of additively manufactured hexagonal and re-entrant (auxetic) lattice structures by using multi-material reinforcements. Additive Manufacturing, 59: 103076 doi: 10.1016/j.addma.2022.103076
[47]	Guo M F, Yang H, Ma L. 2020a. Design and analysis of 2D double-U auxetic honeycombs. Thin-Walled Structures, 155: 106915 doi: 10.1016/j.tws.2020.106915
[48]	Guo M F, Yang H, Ma L. 2020b. Design and characterization of 3D AuxHex lattice structures. International Journal of Mechanical Sciences, 181: 105700 doi: 10.1016/j.ijmecsci.2020.105700
[49]	Guo M F, Yang H, Ma L. 2022. 3D lightweight double arrow-head plate-lattice auxetic structures with enhanced stiffness and energy absorption performance. Composite Structures, 290: 115484 doi: 10.1016/j.compstruct.2022.115484
[50]	Guo Y, Zhang J, Chen L, et al. 2020. Deformation behaviors and energy absorption of auxetic lattice cylindrical structures under axial crushing load. Aerospace Science and Technology, 98: 105662 doi: 10.1016/j.ast.2019.105662
[51]	Hao J, Han D, Zhang X G, et al. 2022. Novel dual-platform lightweight metamaterials with auxeticity. Engineering Structures, 270: 114891 doi: 10.1016/j.engstruct.2022.114891
[52]	He H, Wei X, Yang B, et al. 2022. Ultrastrong and multifunctional aerogels with hyperconnective network of composite polymeric nanofibers. Nature Communications, 13: 4242 doi: 10.1038/s41467-022-31957-2
[53]	Hou Y, Quan J, Thai B Q, et al. 2022. Ultralight biomass-derived carbon fibre aerogels for electromagnetic and acoustic noise mitigation. Journal of Materials Chemistry A, 10: 22771-22780 doi: 10.1039/D2TA06402B
[54]	Hu Q, Lu G, Tse K M. 2024. Dynamic responses of shear thickening fluid-filled lattice structures. International Journal of Impact Engineering, 189: 104954 doi: 10.1016/j.ijimpeng.2024.104954
[55]	Huang J, Gong X, Zhang Q, et al. 2016. In-plane mechanics of a novel zero Poisson's ratio honeycomb core. Composites Part B: Engineering, 89: 67-76 doi: 10.1016/j.compositesb.2015.11.032
[56]	Jiang F, Yang S, Qi C. 2022. Quasi-static crushing response of a novel 3D re-entrant circular auxetic metamaterial. Composite Structures, 300: 116066 doi: 10.1016/j.compstruct.2022.116066
[57]	Jiang W, Zhang X G, Han D, et al. 2023. Experimental and numerical analysis of a novel assembled auxetic structure with two-stage programmable mechanical properties. Thin-Walled Structures, 185: 110555 doi: 10.1016/j.tws.2023.110555
[58]	Jiang Z, Rong J, Chen Z, et al. 2025. Deformation mechanisms and energy absorption characteristics of 3D-printed negative Poisson's ratio sandwich structures subjected to underwater impulsive loading. International Journal of Impact Engineering, 203: 105355 doi: 10.1016/j.ijimpeng.2025.105355
[59]	Jiao C, Yan G. 2021. Design and elastic mechanical response of a novel 3D-printed hexa-chiral helical structure with negative Poisson's ratio. Materials & Design, 212: 110219
[60]	Kashani H, Ito Y, Han J, et al. 2019. Extraordinary tensile strength and ductility of scalable nanoporous graphene. Science Advances, 5: eaat6951 doi: 10.1126/sciadv.aat6951
[61]	Kokkinis D, Schaffner M, Studart A R. 2015. Multimaterial magnetically assisted 3D printing of composite materials. Nature Communications, 6: 8643 doi: 10.1038/ncomms9643
[62]	Lakes R. 1987. Foam structures with a negative Poisson's ratio. Science, 235: 1038-1040 doi: 10.1126/science.235.4792.1038
[63]	Lakes R, Elms K. 1993. Indentability of conventional and negative Poisson's ratio foams. Journal of Composite Materials, 27: 1193-1202 doi: 10.1177/002199839302701203
[64]	Larsen U D, Signund O, Bouwsta S. 1997. Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio. Journal of Microelectromechanical Systems, 6: 99-106 doi: 10.1109/84.585787
[65]	Lendlein A, Gould O E. 2019. Reprogrammable recovery and actuation behaviour of shape-memory polymers. Nature Reviews Materials, 4: 116-133 doi: 10.1038/s41578-018-0078-8
[66]	Li C, Zhou Q, Li H, et al. 2024. Dynamic crushing responses of enhanced auxetic re-entrant honeycomb based on additive manufacturing. Structures, 69: 107367 doi: 10.1016/j.istruc.2024.107367
[67]	Li F, Lin D, Chen Z, et al. 2018. Ultrahigh piezoelectricity in ferroelectric ceramics by design. Nature Materials, 17: 349-354 doi: 10.1038/s41563-018-0034-4
[68]	Li H, Bi K, Han Q, et al. 2025. A state-of-the-art review on negative stiffness-based structural vibration control. Engineering Structures, 323: 119247 doi: 10.1016/j.engstruct.2024.119247
[69]	Li L, He Q, Jing X, et al. 2023. Study on three-point bending behavior of sandwich beams with novel auxetic honeycomb core. Materials Today Communications, 35: 106259 doi: 10.1016/j.mtcomm.2023.106259
[70]	Li Q X, Zhi X D, Fan F. 2022. Quasi-static compressive behaviour of 3D-printed origami-inspired cellular structure: Experimental, numerical and theoretical studies. Virtual and Physical Prototyping, 17: 69-91 doi: 10.1080/17452759.2021.1987051
[71]	Li W, Zhong Y, Zhu Y, et al. 2024. Enhancing the structural stiffness and energy absorption of re-entrant auxetic honeycombs using folded stiffeners. Thin-Walled Structures, 205: 112504 doi: 10.1016/j.tws.2024.112504
[72]	Li X, Fan R, Fan Z, et al. 2021. Programmable mechanical metamaterials based on hierarchical rotating structures. International Journal of Solids and Structures, 216: 145-155 doi: 10.1016/j.ijsolstr.2021.01.028
[73]	Li X, Li Z, Guo Z, et al. 2025. A novel hybrid star honeycomb with individually adjustable second plateau stresses. Composite Structures, 356: 118881 doi: 10.1016/j.compstruct.2025.118881
[74]	Lin H B, Liu H T. 2023. Mechanical properties and band gap characteristics of flexible skin based on multi-concave angle honeycomb. Materials Today Communications, 35: 106113 doi: 10.1016/j.mtcomm.2023.106113
[75]	Lin Y X, Zhong Y F, Hien P L, et al. 2025. Gradient re-entrant honeycomb with quasi-ZPR and improved out-of-plane flexibility through tunable horizontal ligaments. Thin-Walled Structures, 213: 113241 doi: 10.1016/j.tws.2025.113241
[76]	Lira C, Scarpa F, Tai Y H, et al. 2011. Transverse shear modulus of SILICOMB cellular structures. Composites Science and Technology, 71: 1236-1241 doi: 10.1016/j.compscitech.2011.04.008
[77]	Liu H, Deng H, Bao J. 2025. A deployable modular structure with zero thermal expansion for mesh reflector antennas and its structural performance. Structures, 71: 108000 doi: 10.1016/j.istruc.2024.108000
[78]	Liu K, Han L, Hu W, et al. 2020. 4D printed zero Poisson's ratio metamaterial with switching function of mechanical and vibration isolation performance. Materials & Design, 196: 109153
[79]	Liu R, Ji C, Mock J, et al. 2009. Broadband ground-plane cloak. Science, 323: 366-369 doi: 10.1126/science.1166949
[80]	Liu W, Zhu H, Zhou S, et al. 2013. In-plane corrugated cosine honeycomb for 1D morphing skin and its application on variable camber wing. Chinese Journal of Aeronautics, 26: 935-942 doi: 10.1016/j.cja.2013.04.015
[81]	Liu Z, Lv Q, Li D, et al. 2025. A straight-arch-straight beam tandem quasi-zero stiffness structure. International Journal of Mechanical Sciences, 286: 109818 doi: 10.1016/j.ijmecsci.2024.109818
[82]	Lu H, Wang X, Chen T. 2021. In-plane dynamics crushing of a combined auxetic honeycomb with negative Poisson's ratio and enhanced energy absorption. Thin-Walled Structures, 160: 107366 doi: 10.1016/j.tws.2020.107366
[83]	Lu H, Wang X, Chen T. 2022. Design and quasi-static responses of a hierarchical negative Poisson's ratio structure with three plateau stages and three-step deformation. Composite Structures, 291: 115591 doi: 10.1016/j.compstruct.2022.115591
[84]	Lu H, Wang X, Chen T. 2023. Quasi-static bending response and energy absorption of a novel sandwich beam with a reinforced auxetic core under the fixed boundary at both ends. Thin-Walled Structures, 191: 111011 doi: 10.1016/j.tws.2023.111011
[85]	Lu Y, Luo Q, Tong L. 2025. Topology optimization for metastructures with quasi-zero stiffness and snap-through features. Computer Methods in Applied Mechanics and Engineering, 434: 117587 doi: 10.1016/j.cma.2024.117587
[86]	Lu Y, Ma Y, Deng F, et al. 2024. Gradient wood-derived hydrogel actuators constructed by an isotropic-anisotropic structure strategy with rapid thermal-response, high strength and programmable deformation. Chemical Engineering Journal, 504: 158903
[87]	Luo H C, Ren X, Zhang Y, et al. 2022. Mechanical properties of foam-filled hexagonal and re-entrant honeycombs under uniaxial compression. Composite Structures, 280: 114922 doi: 10.1016/j.compstruct.2021.114922
[88]	Maldovan M. 2013. Sound and heat revolutions in phononics. Nature, 503: 209-217 doi: 10.1038/nature12608
[89]	Mao A, Zhao N, Liang Y, et al. 2021. Mechanically efficient cellular materials inspired by cuttlebone. Advanced Materials, 33: 2007348 doi: 10.1002/adma.202007348
[90]	Mirabolghasemi A, Akbarzadeh A, Rodrigue D, et al. 2019. Thermal conductivity of architected cellular metamaterials. Acta Materialia, 174: 61-80 doi: 10.1016/j.actamat.2019.04.061
[91]	Mueller J, Raney J R, Shea K, et al. 2018. Architected lattices with high stiffness and toughness via multicore–shell 3D printing. Advanced Materials, 30: 1705001 doi: 10.1002/adma.201705001
[92]	Muth J T, Dixon P G, Woish L, et al. 2017. Architected cellular ceramics with tailored stiffness via direct foam writing. Proceedings of the National Academy of Sciences, 114: 1832-1837 doi: 10.1073/pnas.1616769114
[93]	Na H, Kang Y W, Park C S, et al. 2022. Hydrogel-based strong and fast actuators by electroosmotic turgor pressure. Science, 376: 301-307 doi: 10.1126/science.abm7862
[94]	Ni X, Wong Z J, Mrejen M, et al. 2015. An ultrathin invisibility skin cloak for visible light. Science, 349: 1310-1314 doi: 10.1126/science.aac9411
[95]	Nicolaou Z G, Motter A E. 2012. Mechanical metamaterials with negative compressibility transitions. Nature Materials, 11: 608-613 doi: 10.1038/nmat3331
[96]	Niu H, Lu J, Qin R, et al. 2025. A self-locked chiral honeycomb: In-plane compression behavior and energy absorption. European Journal of Mechanics-A/Solids, 111: 105580 doi: 10.1016/j.euromechsol.2025.105580
[97]	Ouyang S B, Zhong Y F, Hien P L, et al. 2025. Designing re-entrant nested star-shaped honeycombs for energy-absorbing and load-bearing capabilities. Engineering Structures, 335: 120258 doi: 10.1016/j.engstruct.2025.120258
[98]	Overvelde J T B, Bertoldi K. 2014. Relating pore shape to the non-linear response of periodic elastomeric structures. Journal of the Mechanics and Physics of Solids, 64: 351-366 doi: 10.1016/j.jmps.2013.11.014
[99]	Pan D, Tan S, Zhang Z, et al. 2025. The metastructures actuated by rotational motion with quasi-zero stiffness, negative stiffness, and bistability. Thin-Walled Structures, 207: 112700 doi: 10.1016/j.tws.2024.112700
[100]	Pendry J B, Schurig D, Smith D R. 2006. Controlling electromagnetic fields. Science, 312: 1780-1782 doi: 10.1126/science.1125907
[101]	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
[102]	Qu Y C, Teng X C, Zhang Y, et al. 2025. A novel 3D composite auxetic sandwich panel for energy absorption improvement. Engineering Structures, 322: 119129 doi: 10.1016/j.engstruct.2024.119129
[103]	Rafsanjani A, Pasini D. 2016. Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs. Extreme Mechanics Letters, 9: 291-296 doi: 10.1016/j.eml.2016.09.001
[104]	Ravirala N, Alderson A, Alderson K L. 2007. Interlocking hexagons model for auxetic behaviour. Journal of Materials Science, 42: 7433-7445 doi: 10.1007/s10853-007-1583-0
[105]	Ritchie R O. 2011. The conflicts between strength and toughness. Nature Materials, 10: 817-822 doi: 10.1038/nmat3115
[106]	Sadikbasha S, Pandurangan V. 2023. High velocity impact response of sandwich structures with auxetic tetrachiral cores: Analytical model, finite element simulations and experiments. Composite Structures, 317: 117064 doi: 10.1016/j.compstruct.2023.117064
[107]	Sahariah B J, Baishya M J, Namdeo A, et al. 2023. A novel strategy to design lattice structures with zero Poisson's ratio. Engineering Structures, 288: 116214 doi: 10.1016/j.engstruct.2023.116214
[108]	Schaedler T A, Jacobsen A J, Torrents A, et al. 2011. Ultralight metallic microlattices. Science, 334: 962-965 doi: 10.1126/science.1211649
[109]	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
[110]	Schurig D, Mock J J, Justice B J, et al. 2006. Metamaterial electromagnetic cloak at microwave frequencies. Science, 314: 977-980 doi: 10.1126/science.1133628
[111]	Schwartz J, Boydston A. 2019. Multimaterial actinic spatial control 3D and 4D printing. Nature Communications, 10: 791 doi: 10.1038/s41467-019-08639-7
[112]	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
[113]	Sigmund O, Torquato S, Aksay I A. 1998. On the design of 1-3 piezocomposites using topology optimization. Journal of Materials Research, 13: 1038-1048 doi: 10.1557/JMR.1998.0145
[114]	Sklan S R, Li B. 2018. Thermal metamaterials: Functions and prospects. National Science Review, 5: 138-141 doi: 10.1093/nsr/nwy005
[115]	Smith C W, Grima J N, Evans K E. 2000. A novel mechanism for generating auxetic behaviour in reticulated foams: Missing rib foam model. Acta Materialia, 48: 4349-4356 doi: 10.1016/S1359-6454(00)00269-X
[116]	Song Z, Guo D, Liu Y, et al. 2025. Design of kirigami metamaterials with square-symmetric auxeticity under large stretching. Thin-Walled Structures, 213: 113268 doi: 10.1016/j.tws.2025.113268
[117]	Sydney Gladman A, Matsumoto E A, Nuzzo R G, et al. 2016. Biomimetic 4D printing. Nature Materials, 15: 413-418 doi: 10.1038/nmat4544
[118]	Tan X, Chu K, Chen Z, et al. 2024. Recent advances in self-healing hydrogel composites for flexible wearable electronic devices. Nano Research Energy, 3: e9120123 doi: 10.26599/NRE.2024.9120123
[119]	Tang Y X, Zhong Y F, Zhu Y L, et al. 2025. Energy absorption characteristics and auxetic effect of novel elliptic-arc re-entrant honeycomb structures. Engineering Structures, 323: 119260 doi: 10.1016/j.engstruct.2024.119260
[120]	Theocaris P, Stavroulakis G, Panagiotopoulos P. 1997. Negative Poisson's ratios in composites with star-shaped inclusions: A numerical homogenization approach. Archive of Applied Mechanics, 67: 274-286 doi: 10.1007/s004190050117
[121]	Toombs J T, Luitz M, Cook C C, et al. 2022. Volumetric additive manufacturing of silica glass with microscale computed axial lithography. Science, 376: 308-312 doi: 10.1126/science.abm6459
[122]	Virk K, Monti A, Trehard T, et al. 2013. SILICOMB PEEK Kirigami cellular structures: Mechanical response and energy dissipation through zero and negative stiffness. Smart Materials and Structures, 22: 084014 doi: 10.1088/0964-1726/22/8/084014
[123]	Wang H, Lu Z, Yang Z, et al. 2019. A novel re-entrant auxetic honeycomb with enhanced in-plane impact resistance. Composite Structures, 208: 758-770 doi: 10.1016/j.compstruct.2018.10.024
[124]	Wang J, Luo X, Wang K, et al. 2022. On impact behaviors of 3D concave structures with negative Poisson's ratio. Composite Structures, 298: 115999 doi: 10.1016/j.compstruct.2022.115999
[125]	Wang N, Liu W, Tang A, et al. 2014. Strain isolation: A simple mechanism for understanding and detecting structures of zero Poisson's ratio. Physica Status Solidi B, 251: 2239-2246 doi: 10.1002/pssb.201451376
[126]	Wang Q, Tian X, Huang L, et al. 2018. Programmable morphing composites with embedded continuous fibers by 4D printing. Materials & Design, 155: 404-413
[127]	Wang S, Liu H T. 2023. Energy absorption performance of the auxetic arc-curved honeycomb with thickness and arc angle gradient based on additive manufacturing. Materials Today Communications, 35: 105515 doi: 10.1016/j.mtcomm.2023.105515
[128]	Wang S, Liu H T. 2024. Quasi-static compression response of a novel multi-step auxetic honeycomb with tunable transition strain. Aerospace Science and Technology, 155: 109730 doi: 10.1016/j.ast.2024.109730
[129]	Wang S, Liu H T, Cai G B. 2024. Programmable mechanical responses of a hybrid star-rhombus honeycomb based on digital design method. Thin-Walled Structures, 205: 112399 doi: 10.1016/j.tws.2024.112399
[130]	Wang W J, Yang H, Zhang W M, et al. 2025. Experimental study on the impact resistance of fill-enhanced mechanical metamaterials. International Journal of Mechanical Sciences, 285: 109799 doi: 10.1016/j.ijmecsci.2024.109799
[131]	Wang W J, Zhang W M, Guo M F, et al. 2023. Energy absorption characteristics of a lightweight auxetic honeycomb under low-velocity impact loading. Thin-Walled Structures, 185: 110577 doi: 10.1016/j.tws.2023.110577
[132]	Wang W J, Zhang W M, Guo M F, et al. 2024. Impact resistance of assembled plate-lattice auxetic structures. Composite Structures, 338: 118132 doi: 10.1016/j.compstruct.2024.118132
[133]	Wang X T, Wang B, Wen Z H, et al. 2018. Fabrication and mechanical properties of CFRP composite three-dimensional double-arrow-head auxetic structures. Composites Science and Technology, 164: 92-102 doi: 10.1016/j.compscitech.2018.05.014
[134]	Wang Y C, Lakes R. 2005. Composites with inclusions of negative bulk modulus: Extreme damping and negative Poisson's ratio. Journal of Composite Materials, 39: 1645-1657 doi: 10.1177/0021998305051112
[135]	Wei T, Lu F, Zhang C, et al. 2025. Energy absorption of 3D assembled auxetic meta-structure with compression-twisting effect. Structures, 73: 108482 doi: 10.1016/j.istruc.2025.108482
[136]	Wojciechowski K W. 1989. Two-dimensional isotropic system with a negative Poisson ratio. Physics Letters A, 137: 60-64 doi: 10.1016/0375-9601(89)90971-7
[137]	Wu L, Zhao F, Lu Z, et al. 2022. Impact energy absorption composites with shear stiffening gel-filled negative Poisson's ratio skeleton by kirigami method. Composite Structures, 298: 116009 doi: 10.1016/j.compstruct.2022.116009
[138]	Wu W, Hu W, Qian G, et al. 2019. Mechanical design and multifunctional applications of chiral mechanical metamaterials: A review. Materials & Design, 180: 107950
[139]	Wu X, Su Y, Shi J. 2020. In-plane impact resistance enhancement with a graded cell-wall angle design for auxetic metamaterials. Composite Structures, 247: 112451 doi: 10.1016/j.compstruct.2020.112451
[140]	Wu X, Zhang S, Ding L, et al. 2024. Bi-material multistable auxetic honeycombs with reusable and enhanced energy-absorbing phases under in-plane crushing. Thin-Walled Structures, 201: 111988 doi: 10.1016/j.tws.2024.111988
[141]	Xia X, Afshar A, Yang H, et al. 2019. Electrochemically reconfigurable architected materials. Nature, 573: 205-213 doi: 10.1038/s41586-019-1538-z
[142]	Xu C, Stiubianu G T, Gorodetsky A A. 2018. Adaptive infrared-reflecting systems inspired by cephalopods. Science, 359: 1495-1500 doi: 10.1126/science.aar5191
[143]	Xu H, Liu H T, Li G F. 2025. In-plane characteristics of a multi-arc re-entrant auxetic honeycomb with enhanced negative Poisson's ratio effect and energy absorption. European Journal of Mechanics-A/Solids, 109: 105473 doi: 10.1016/j.euromechsol.2024.105473
[144]	Xu M, Xu Z, Zhang Z, et al. 2019. Mechanical properties and energy absorption capability of AuxHex structure under in-plane compression: Theoretical and experimental studies. International Journal of Mechanical Sciences, 159: 43-57 doi: 10.1016/j.ijmecsci.2019.05.044
[145]	Yan Y, Xu S, Wang X, et al. 2025. A photothermal-responsive and glucose-responsive antibacterial hydrogel featuring tunable mechanical properties. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 708: 136029 doi: 10.1016/j.colsurfa.2024.136029
[146]	Yang H, D'Ambrosio N, Liu P, et al. 2023. Shape memory mechanical metamaterials. Materials Today, 66: 36-49 doi: 10.1016/j.mattod.2023.04.003
[147]	Yang H, Ma L. 2021. Impact resistance of additively manufactured 3D double-U auxetic structures. Thin-Walled Structures, 169: 108373 doi: 10.1016/j.tws.2021.108373
[148]	Yang H, Yang L, Zheng X, et al. 2025. High-performance 3D auxetic metamaterials enabled by multiple auxetic mechanisms. International Journal of Mechanical Sciences, 287: 109981 doi: 10.1016/j.ijmecsci.2025.109981
[149]	Yang S, Chalivendra V B, Kim Y K. 2017. Fracture and impact characterization of novel auxetic Kevlar®/Epoxy laminated composites. Composite Structures, 168: 120-129 doi: 10.1016/j.compstruct.2017.02.034
[150]	Yeo S J, Oh M J, Yoo P J. 2019. Structurally controlled cellular architectures for high-performance ultra-lightweight materials. Advanced Materials, 31: 1803670 doi: 10.1002/adma.201803670
[151]	Yu P, Zhang P, Ji Q, et al. 2024. A multi-step auxetic metamaterial with instability regulation. International Journal of Solids and Structures, 305: 113040 doi: 10.1016/j.ijsolstr.2024.113040
[152]	Yu R, Luo W, Yuan H, et al. 2020. Experimental and numerical research on foam filled re-entrant cellular structure with negative Poisson's ratio. Thin-Walled Structures, 153: 106679 doi: 10.1016/j.tws.2020.106679
[153]	Yu X, Chen H, Lin H, et al. 2014. Continuously tuning effective refractive index based on thermally controllable magnetic metamaterials. Optics Letters, 39: 4643-4646 doi: 10.1364/OL.39.004643
[154]	Yu X, Zhou J, Liang H, et al. 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
[155]	Yuan C, Mu X, Dunn C K, et al. 2018. Thermomechanically triggered two-stage pattern switching of 2D lattices for adaptive structures. Advanced Functional Materials, 28: 1705727 doi: 10.1002/adfm.201705727
[156]	Yue L, Liu H, Cheng Z, et al. 2024. Dynamic crushing behavior of a novel bi-directional gradient lattice structure under axial and oblique impact loadings. Thin-Walled Structures, 198: 111697 doi: 10.1016/j.tws.2024.111697
[157]	Ze Q, Kuang X, Wu S, et al. 2020. Magnetic shape memory polymers with integrated multifunctional shape manipulation. Advanced Materials, 32: 1906657 doi: 10.1002/adma.201906657
[158]	Zhang B, Zhang W, Zhang Z, et al. 2019. Self-healing four-dimensional printing with an ultraviolet curable double-network shape memory polymer system. ACS Applied Materials & Interfaces, 11: 10328-10336
[159]	Zhang C, Lu F, Mo W, et al. 2025. Dynamic responses and energy absorption characteristics of windmill-shaped auxetic structure under impact loading. Structures, 75: 108670 doi: 10.1016/j.istruc.2025.108670
[160]	Zhang D, Lim X J G, Li X, et al. 2022. 3D-Printed porous thermoelectrics for in situ energy harvesting. ACS Energy Letters, 8: 332-338
[161]	Zhang H, Chen P, Lin G, et al. 2022. A corrugated gradient mechanical metamaterial: Lightweight, tunable auxeticity and enhanced specific energy absorption. Thin-Walled Structures, 176: 109355 doi: 10.1016/j.tws.2022.109355
[162]	Zhang Q, Sun Y. 2025. Energy absorption characteristic of auxetic metamaterials honeycombs and lattices with negative thermal expansion. Thin-Walled Structures, 208: 112824 doi: 10.1016/j.tws.2024.112824
[163]	Zhang W, Chen J, Li X, et al. 2020. Liquid metal-polymer microlattice metamaterials with high fracture toughness and damage recoverability. Small, 16: 2004190 doi: 10.1002/smll.202004190
[164]	Zhang W, Wang H, Lou X, et al. 2024. On in-plane crushing behavior of a combined re-entrant double-arrow honeycomb. Thin-Walled Structures, 194: 111303 doi: 10.1016/j.tws.2023.111303
[165]	Zhang X, Vyatskikh A, Gao H, et al. 2019. Lightweight, flaw-tolerant, and ultrastrong nanoarchitected carbon. Proceedings of the National Academy of Sciences, 116: 6665-6672 doi: 10.1073/pnas.1817309116
[166]	Zhang Y, Ren X, Jiang W, et al. 2022. In-plane compressive properties of assembled auxetic chiral honeycomb composed of slotted wave plate. Materials & Design, 221: 110956
[167]	Zhang Z, Gu Y W, Wu H A, et al. 2025. Investigation on the energy absorption characteristics of novel graded auxetic re-entrant honeycombs. Composite Structures, 352: 118633 doi: 10.1016/j.compstruct.2024.118633
[168]	Zhang Z, Lei Y P, Wang H. 2025. Deformation and energy absorption characteristics of graded auxetic metamaterials featuring peanut-shaped perforations under in-plane compression. International Journal of Solids and Structures, 313: 113318 doi: 10.1016/j.ijsolstr.2025.113318
[169]	Zheng X, Lee H, Weisgraber T H, et al. 2014. Ultralight, ultrastiff mechanical metamaterials. Science, 344: 1373-1377 doi: 10.1126/science.1252291
[170]	Zhou C, Zhang F, Zhang X, et al. 2025. Hierarchical negative stiffness structures with improved resilience and energy absorption capability. Materials Today Communications, 42: 111371 doi: 10.1016/j.mtcomm.2024.111371
[171]	Zhou J, Wang Y, Luo H, et al. 2024. Energy absorption of auxetic honeycomb with graded beam thickness based on Bezier curve. Aerospace Science and Technology, 155: 109619 doi: 10.1016/j.ast.2024.109619
[172]	Zhou Y, Chen C, Zhu S, et al. 2019. A printed, recyclable, ultra-strong, and ultra-tough graphite structural material. Materials Today, 30: 17-25 doi: 10.1016/j.mattod.2019.03.016
[173]	Zhu Y, Fu Y, Rui X, et al. 2025. On the design and crashworthiness of a novel auxetic self-locking energy absorption system. International Journal of Solids and Structures, 311: 113246 doi: 10.1016/j.ijsolstr.2025.113246
[174]	Zouaoui M, Saifouni O, Gardan J, et al. 2022. Improvement of fracture toughness based on auxetic patterns fabricated by metallic extrusion in 3D printing. Procedia Structural Integrity, 42: 680-686 doi: 10.1016/j.prostr.2022.12.086