Abstract
A novel hierarchical graded structure was designed based on scaling the hierarchical re-entrant cells, and its quasi-static compressive response was experimentally investigated and compared with a similar non-graded structure. The specimens were fabricated using fused deposition modeling (FDM) with Acrylonitrile Butadiene Styrene (ABS) material. The new design significantly improves the energy absorption characteristics of the structure, such as enhanced specific energy absorption up to 290% compared to the non-graded structure. The mechanical response and progressive damage of the introduced structure were successfully simulated using finite element analysis. Instead of a popular elastic–plastic model, a pressure-dependent plasticity material model, specifically the Drucker-Prager model, was employed. Verification of the results with experimental data confirms the adequacy of this model for accurately simulating the examined auxetic structure.
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References
Yang OHL, West H, Cormier D (2015) Mechanical properties of 3D re-entrant honeycomb auxetic structures realized via additive manufacturing. Int J Solids Struct 69–70:475–490. https://doi.org/10.1016/j.ijsolstr.2015.05.005
Mahshid HNHR, Loft Hojbjerre K (2016) Strength analysis and modeling of cellular lattice structures manufactured using selective laser melting for tooling applications. Mater Des 104:276–283. https://doi.org/10.1016/j.matdes.2016.05.020
Bertocco LEA, Aurino A, Borrelli D, Caraviello A (2021) Influence of SLM parameters on the compressive behaviour of lattice structures in 17-4PH stainless steel. IOP Conf Ser Mater Sci Eng 1038(1):012035
Zhang GLJ, You Z (2020) Large deformation and energy absorption of additively manufactured auxetic materials and structures: a review. Compos Part B 201:108340. https://doi.org/10.1016/j.compositesb.2020.108340
Gunaydın GSK, Turkmen HS, Grande AM (2022) Failure analysis of auxetic lattice structures under crush load. Proc Struct Int 35:237–246
Ha GLNS (2020) Thin-walled corrugated structures: a review of crashworthiness designs and energy absorption characteristics. Thin Walled Struct 157:106995. https://doi.org/10.1016/j.tws.2020.106995
Ren RDX, Tran P, Ngo TD, **e YM (2018) Auxetic metamaterials and structures: a review. Smart Mater Struct 27:023001. https://doi.org/10.1088/1361-665X/aaa61c
Jiang XRW, Wang SL, Zhang XG, Zhang XY, Luo C, **e YM, Scarpa F, Alderson A, Evans KE (2022) Manufacturing, characteristics and applications of auxetic foams: a state-of-the-art review. Compos Part B 235:1–27. https://doi.org/10.1016/j.compositesb.2022.109733
** ZWX, Ning J, **ao G, Liu E, Shu X (2016) Dynamic response of sandwich structures with graded auxetic honeycomb cores under blast loading. Compos Part B 106:206–217. https://doi.org/10.1016/j.compositesb.2016.09.037
Najafi HAM, Liaghat Gh (2021) Experimental investigation on energy absorption of auxetic structures. Mater Today Proc 34:350–355. https://doi.org/10.1016/j.matpr.2020.06.075
Zhang HHX, Tian R, Xue Q, Guan H, Yang X (2022) Static compression and dynamic crushing behaviors of novel hybrid re-entrant auxetic metamaterials with enhanced energy-absorption. Compos Struct 288:115399. https://doi.org/10.1016/j.compstruct.2022.115399
Tan ZCHHL, Li E, Tan XW, Cheng AG, Li QQ (2020) Energy absorption characteristics of three-layered sandwich panels with graded re-entrant hierarchical honeycombs cores. Aerosp Sci Technol 106:106073. https://doi.org/10.1016/j.ast.2020.106073
Zhang HLET, Ng BF (2021) Novel arc-shaped ligaments to enhance energy absorption capabilities of re-entrant anti-trichiral structures. Compos Part B 227:109366. https://doi.org/10.1016/j.compositesb.2021.109366
Liu NWW, Luo T, Lin Z (2016) In-plane dynamic crushing of re-entrant auxetic cellular structure. Mater Des 100:84–91. https://doi.org/10.1016/j.matdes.2016.03.086
Wu WHW, Qian G, Liao H, Xu X, Berto F (2019) Mechanical design and multifunctional applications of chiral mechanical metamaterials: a review. Mater Des 180:107950. https://doi.org/10.1016/j.matdes.2019.107950
Qiao CQCJ (2015) Analyses on the in-plane impact resistance of auxetic double arrowhead honeycombs. J Appl Mech 82:051007. https://doi.org/10.1115/1.4030007
Wei XZL, Yu Q, Zhang W, Zhu G (2021) In-plane compression behaviors of the auxetic star honeycomb: experimental and numerical simulation. Aerosp Sci Technol 115:106797. https://doi.org/10.1016/j.ast.2021.106797
Linforth TNS, Tran P, Ruan D, Odish R (2021) Investigation of the auxetic oval structure for energy absorption through quasi-static and dynamic experiments. Int J Impact Eng 147:1–13. https://doi.org/10.1016/j.ijimpeng.2020.103741
Ebrahimi RHMS, Etemadi E (2022) In-plane energy absorption characteristics and mechanical properties of 3D printed novel hybrid cellular structures. J Mater Res Technol 20:3616–3632. https://doi.org/10.1016/j.jmrt.2022.08.064
Choudhry BPNK, Kumar S (2022) In-plane energy absorption characteristics of a modified re-entrant auxetic structure fabricated via 3D printing. Compos Part B 228:109437. https://doi.org/10.1016/j.compositesb.2021.109437
Zhang PCH, Lin G, Sun W (2022) A corrugated gradient mechanical metamaterial: Lightweight, tunable auxeticity and enhanced specific energy absorption. Thin Walled Struct 176:109355. https://doi.org/10.1016/j.tws.2022.109355
Yang CHL, Wu H, Hao L, Wei Q, Yan Ch, Shi Y (2020) Insights into unit cell size effect on mechanical responses and energy absorption capability of titanium graded porous structures manufactured by laser powder bed fusion. J Mech Behav Biomed Mater 109:103843. https://doi.org/10.1016/j.jmbbm.2020.103843
Xu XZF, Zhang H (2018) A review on functionally graded structures and materials for energy absorption. Eng Struct 171:309–325. https://doi.org/10.1016/j.engstruct.2018.05.094
Niknam AHAH (2020) Graded lattice structures: simultaneous enhancement in stiffness and energy absorption. Mater Des 196:109129. https://doi.org/10.1016/j.matdes.2020.109129
Anni KZUIA, Pagliocca N, Singh N, Rahman O, Youssef G, Koohbor B (2022) Out-of-plane load-bearing and mechanical energy absorption properties of flexible density-graded TPU honeycombs. Compos Part C Open Access 8:100284. https://doi.org/10.1016/j.jcomc.2022.100284
Yang RML, Ferrucci M, Yan C, Shi Y, Yang S (2019) Continuous graded gyroid cellular structures fabricated by selective laser melting: design, manufacturing and mechanical properties. Mater Des 162:394–404. https://doi.org/10.1016/j.matdes.2018.12.007
Li XZQ, Fan F (2022) Dynamic crushing of uniform and functionally graded origami-inspired cellular structure fabricated by SLM. Eng Struct 262:114327. https://doi.org/10.1016/j.engstruct.2022.114327
**ang ZFX, Zhang Sh, Lu G, Ha N, Liang Y, Zhang X (2021) The mechanical characteristics of graded Miura-ori metamaterials. Mater Des 211:110173. https://doi.org/10.1016/j.matdes.2021.110173
Wu X, Su Y, Shi J (2020) In-plane impact resistance enhancement with a graded cell-wall angle design for auxetic metamaterials. Compos Struct 247:112451. https://doi.org/10.1016/j.compstruct.2020.112451
**ao ZDD, Li Y, Wu W, Fang D (2019) Compression behavior of the graded metallic auxetic reentrant honeycomb: experiment and finite element analysis. Mater Sci Eng A 758:163–171. https://doi.org/10.1016/j.msea.2019.04.116
Duraibabu RPRV, Sugavaneswaran M, Arumaikkannu G (2020) Compression behavior of Functionally graded cellular materials fabricated with FDM. Mater Today Proc 24(2):1035–1041. https://doi.org/10.1016/j.matpr.2020.04.417
Mohammadi AZKM, Bodaghi M, Long J, Khoo SY, **ang Y, Zolfagharian A (2023) Sustainable Robotic Joints 4D Printing with Variable Stiffness Using Reinforcement Learning. Robot Comput Integr Manuf 85:102636. https://doi.org/10.1016/j.rcim.2023.102636
Qi QLD, He Ch, Li Y, Wu W, **ao D (2019) Impact energy absorption of functionally graded chiral honeycomb structures. Extreme Mech Lett 32:100568. https://doi.org/10.1016/j.eml.2019.100568
Novak LON, Ren Z, Vesenjak M (2020) Mechanical properties of hybrid metamaterial with auxetic chiral cellular structure and silicon filler. Compos Struct 234:111718. https://doi.org/10.1016/j.compstruct.2019.111718
Wang FLY, Zhang X, Zhang K, Wang X, Gan D, Yang B (2021) Cell-size graded sandwich enhances additive manufacturing fidelity and energy absorption. Int J Mech Sci 211:106798. https://doi.org/10.1016/j.ijmecsci.2021.106798
Plocher APJ (2020) Effect of density and unit cell size grading on the stiffness and energy absorption of short fibre-reinforced functionally graded lattice structures. Addit Manuf 33:101171. https://doi.org/10.1016/j.addma.2020.101171
Novak MBN, Al-Ketan O, Ren Z, Vesenjak M (2022) Impact and blast resistance of uniform and graded sandwich panels with TPMS cellular structures. Compos Struct 300:116174. https://doi.org/10.1016/j.compstruct.2022.116174
Liu ETZH, Wang G, Ng BF (2022) In-plane crushing behavior and energy absorption of a novel graded honeycomb from hierarchical architecture. Int J Mech Sci 221:107202. https://doi.org/10.1016/j.ijmecsci.2022.107202
Wang JZZh, Li Zh, Shi Ch (2020) On the crashworthiness of bio-inspired hexagonal prismatic tubes under axial compression. Int J Mech Sci 186:105893. https://doi.org/10.1016/j.ijmecsci.2020.105893
Keripale SBKA (2022) Analysis of impact energy absorption and dissipation capacity of different composite materials using numerical simulation. Mater Today Proc 59:661–666. https://doi.org/10.1016/j.matpr.2021.12.196
Rahman BKO (2020) Optimization of energy absorption performance of polymer honeycombs by density gradation. Compos Part C Open Access 3:100052. https://doi.org/10.1016/j.jcomc.2020.100052
Choudhry SRBNK, Panda B, Singh H (2022) Experimental and numerical analysis of the bending behavior of 3D printed modified auxetic sandwich structures. Mater Today Proc 56(3):1356–1363. https://doi.org/10.1016/j.matpr.2021.11.425
Zhang HHX, Tian R, Xue Q, Guan H, Yang X (2022) Quasi-static compression and dynamic crushing behaviors of novel hybrid re-entrant auxetic metamaterials with enhanced energy-absorption. Compos Struct 288:115399. https://doi.org/10.1016/j.compstruct.2022.115399
Zhou BFNJ, Han N, Xu Sh, Zou M (2023) Crashworthiness and optimization of bionic sandwich cores under out-of-plane compression. Int J Mech Sci 246:108137. https://doi.org/10.1016/j.ijmecsci.2023.108137
Wang WZW, Guo M, Yang J, Ma L (2023) Energy absorption characteristics of a lightweight auxetic honeycomb under low-velocity impact loading. Thin Walled Struct 185:110577. https://doi.org/10.1016/j.tws.2023.110577
Kim PNY, Kim H, Choi Y (2022) Multi-morphology cellular structure design with smooth transition of geometry and homogenized mechanical properties between adjacent cells. Mater Des 218:110727. https://doi.org/10.1016/j.matdes.2022.110727
Syaefudin EA, Kholil A, Hakim M, Wulandari DA, Riyadi, Murtinugraha E (2023) The effect of orientation on tensile strength 3D printing with ABS and PLA materials. J Phys Conf Ser 2596:012002. https://doi.org/10.1088/1742-6596/2596/1/012002
Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Quart Appl Math 10:157–165. https://doi.org/10.1090/qam/48291
Zhang ZLW, Yang J, Ma L, Lin Z, Schmidt R, Schroder K (2022) A lightweight rotationally arranged auxetic structure with excellent energy absorption performance. Mech Mater 166:104244. https://doi.org/10.1016/j.mechmat.2022.104244
Dong YLZ, Zhao T, Wu W, **ao D, Liang J (2019) Experimental and numerical studies on the compressive mechanical properties of the metallic auxetic reentrant honeycomb. Mater Des 182:108036. https://doi.org/10.1016/j.matdes.2019.108036
Li IMQM, Harrigan JJ (2006) Compressive strain at the onset of the densification of cellular solids. J Cell Plast. https://doi.org/10.1177/0021955X06063519
J. Blaber. "http://www.ncorr.com." (accessed 2018).
Namvar IMN, Zolfagharian A, Demoly F, Bodaghi M (2023) Bio-inspired design, modeling, and 3D printing of lattice-based scale model scooter decks. Int J Adv Manuf Technol 126:2887–2903. https://doi.org/10.1007/s00170-023-11185-8
Zolfagharian PPA, Bodaghi M, Fard M, Rolfe B (2023) Additive manufacturing of composite foam metamaterial springs for vibration isolation. Adv Eng Mater 25:2300356. https://doi.org/10.1002/adem.202300356
Eryildiz M (2021) Effect of build orientation on mechanical behaviour and build time of FDM 3D-printed PLA parts: an experimental investigation. Eur Mech Sci 5:116–120. https://doi.org/10.26701/ems.881254
Tolochyna NZ-LO, Podrezov Y, Verbylo D, Tolochyn O, Zgalat-Lozynskyy O (2023) The role of flexible polymer composite materials properties in energy absorption of three-dimensional auxetic lattice structures. Materi Today Commun 37:107370. https://doi.org/10.1016/j.mtcomm.2023.107370
Wu JFY, Wu Ch, Li C, Sun G, Li Q (2023) Additively manufactured materials and structures: a state-of-the-art review on their mechanical characteristics and energy absorption. Int J Mech Sci 246:108102. https://doi.org/10.1016/j.ijmecsci.2023.108102
Sun DCG, Zhu G, Li Q (2022) Lightweight hybrid materials and structures for energy absorption: a state-of-the-art review and outlook. Thin Walled Struct 172:108760. https://doi.org/10.1016/j.tws.2021.108760
Riot EPA, Cosculluela A, Montemurro M (2023) Influence of manufacturing process-induced geometrical defects on the energy absorption capacity of polymer lattice structures. Def Technol. https://doi.org/10.1016/j.dt.2023.09.003
Shi FTL, Luo Y (2023) Energy absorption characteristics of the bionic lotus petiole structure under transverse load. Thin Walled Struct 187:110748. https://doi.org/10.1016/j.tws.2023.110748
Fan RXW, Davidson M, Yin H, Lai K, Wu Q (2023) Crashworthiness and energy absorption of UHPFRC-steel composite sandwich structures under impact loading. Compos Struct 311:116813. https://doi.org/10.1016/j.compstruct.2023.116813
Nam MJR, Niknam H, Rahmat M, Ashrafi B, Naguib HE (2023) 3D printed octet plate-lattices for tunable energy absorption. Mater Des 228:111835. https://doi.org/10.1016/j.matdes.2023.111835
Rodrigo SXCh, Durandet Y, Tran Ph, Ruan D (2023) Mechanical response of functionally graded lattices with different density grading strategies. Thin Walled Struct 192:111132. https://doi.org/10.1016/j.tws.2023.111132
Rodrigo SXCh, Durandet Y, Fraser D, Ruan D (2023) Quasi-static and dynamic compression of additively manufactured functionally graded lattices: experiments and simulations. Eng Struct 284:115909. https://doi.org/10.1016/j.engstruct.2023.115909
Sang WWL, Sun Zh, Wang F, Xu J, Tian J, Zhao Y, Zhang H (2023) Reusability and energy absorption behavior of 4D-printed heterogeneous lattice structures based on biomass shape memory polyester. J Mater Res Technol 27:1563–1578. https://doi.org/10.1016/j.jmrt.2023.09.323
Novak HA-RN, Airoldi A, Krstulović-Opara L, Łodygowski T, Ren Z, Vesenjak M (2023) Quasi-static and impact behaviour of foam-filled graded auxetic panel. Int J Impact Eng 178:104606. https://doi.org/10.1016/j.ijimpeng.2023.104606
Hu KLK, Gu D, Yang J, Wang H, Yuan L (2019) Mechanical properties and deformation behavior under compressive loading of selective laser melting processed bio-inspired sandwich structures. Mater Sci Eng A 762:138089. https://doi.org/10.1016/j.msea.2019.138089
Soleimanzadeh BRH, Bodaghi M, Jamalabadi M, Zhang AZX (2023) Sustainable Robots 4D Printing. Adv Sust Syst 7:2300289. https://doi.org/10.1002/adsu.202300289
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Mortezapour, A.R., Monazzah, A.H., Sarfaraz, R. et al. Energy absorption of multi-scale hierarchically graded auxetic structures: experimental and simulation methods. Prog Addit Manuf (2024). https://doi.org/10.1007/s40964-024-00623-y
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DOI: https://doi.org/10.1007/s40964-024-00623-y