A review of additively manufactured pentamode lattices: Theory, methods and applications
DOI:
https://doi.org/10.15282/jmes.20.2.2026.8.0876Keywords:
Additive manufacturing, Lattice structures, Metamaterial, pentamodeAbstract
Pentamode metamaterials, characterized by their near-zero shear modulus and high bulk modulus, have attracted solid interest due to their unique ability to mimic fluid-like behaviour. Pentamode metamaterials have driven by improvements in computational design, material selection and advanced manufacturing techniques, especially additive manufacturing. As a comprehensive review article specificially addressing pentamode metamaterials produced by additive manufacturing, this review systematically represents the theoretical foundations, essential design concepts, and main parameters, including the high bulk-to-shear modulus ratio that presents fluidic behaviour. Recent advancements in manufacturing methods such as selective laser melting, direct laser writing, and hybrid fabrication processes are also discussed, along with their impact on achieving precise lattice geometries. Extensive analysis of acoustic and mechanical applications reveals pentamode metamaterials’ versatility in acoustic cloaking, wave scattering reduction, vibration isolation, shock absorption and tunable stiffness devices. Despite intensive progress, challenges such as scalability, precision of manufacturing, and material optimization continue to emerge. This systematic review concludes by highlighting future research opportunities, emphasizing hybrid material usage, computational optimization, and real-world applications, enhancing the transformative potential of pentamode metamaterials in acoustic and mechanical applications.
References
[1] G. W. Milton and A. V. Cherkaev, "Which elasticity tensors are realizable?" Journal of Engineering Materials and Technology, vol. 117, no. 4, pp. 483–493, 1995. https://doi.org/10.1115/1.2804743
[2] E. Barchiesi, M. Spagnuolo, and L. Placidi, "Mechanical metamaterials: A state of the art," Mathematics and Mechanics of Solids, vol. 24, no. 1, pp. 212–234, 2019. https://doi.org/10.1177/1081286517735695
[3] Y. Guo, T. Dekorsy, and M. Hettich, "Topological guiding of elastic waves in phononic metamaterials based on 2D pentamode structures," Scientific Reports, vol. 7, no. 1, Art. no. 18043, 2017. https://doi.org/10.1038/s41598-017-18394-8
[4] T. Bückmann, M. Thiel, M. Kadic, R. Schittny, and M. Wegener, “An elasto-mechanical unfeelability cloak made of pentamode metamaterials,” Nature Communications, vol. 5, no. 1, Art. no. 4130, 2014. https://doi.org/10.1038/ncomms5130
[5] G. C. Mendez, J. M. Podesta, O. Valls-Lloberas, S. Toro, A. E. Huespe, and J. Oliver, “Computational material design for acoustic cloaking,” International Journal for Numerical Methods in Engineering, vol. 112, no. 10, pp. 1353–1380, 2017. https://doi.org/10.1002/nme.5560
[6] C. Cai, Z. Wang, Y. Chu, G. Liu, and Z. Xu, “The phononic band gaps of Bragg scattering and locally resonant pentamode metamaterials,” Journal of Physics D: Applied Physics, vol. 50, no. 41, Art. no. 415302, 2017. https://doi.org/10.1088/1361-6463/aa83ec
[7] F. Fraternali and A. Amendola, “Mechanical modeling of innovative metamaterials alternating pentamode lattices and confinement plates,” Journal of the Mechanics and Physics of Solids, vol. 99, pp. 259–271, 2017. https://doi.org/10.1016/j.jmps.2016.11.010
[8] A. Amendola, G. Carpentieri, L. Feo, and F. Fraternali, “Bending dominated response of layered mechanical metamaterials alternating pentamode lattices and confinement plates,” Composite Structures, vol. 157, pp. 71–77, 2016. https://doi.org/10.1016/j.compstruct.2016.07.031
[9] C. L. Scandrett, J. E. Boisvert, and T. R. Howarth, “Broadband optimization of a pentamode-layered spherical acoustic waveguide,” Wave Motion, vol. 48, no. 6, pp. 505–514, 2011. https://doi.org/10.1016/j.wavemoti.2011.02.007
[10] M. Kadic, T. Bückmann, N. Stenger, M. Thiel, and M. Wegener, “On the practicability of pentamode mechanical metamaterials,” Applied Physics Letters, vol. 100, no. 19, Art. no. 191901, 2012. https://doi.org/10.1063/1.4709436
[11] G. F. Mejica and A. D. Lantada, “Comparative study of potential pentamodal metamaterials inspired by Bravais lattices,” Smart Materials and Structures, vol. 22, no. 11, Art. no. 115013, 2013. https://doi.org/10.1088/0964-1726/22/11/115013
[12] Z. Wang, G. Liu, C. Cai, Y. Chu, Y. Li, and Z. Xu, “Connection overlap deviation analysis of pentamode metamaterials,” Journal of Physics D: Applied Physics, vol. 50, no. 48, Art. no. 485302, 2017. https://doi.org/10.1088/1361-6463/aa92a6
[13] Y. Huang, X. Lu, G. Liang, and Z. Xu, “Pentamodal property and acoustic band gaps of pentamode metamaterials with different cross-section shapes,” Physics Letters A, vol. 380, no. 13, pp. 1334–1338, 2016. https://doi.org/10.1016/j.physleta.2016.01.041
[14] F. Calignano and V. Mercurio, “An overview of the impact of additive manufacturing on supply chain, reshoring, and sustainability,” Cleaner Logistics and Supply Chain, vol. 7, Art. no. 100103, 2023. https://doi.org/10.1016/j.clscn.2023.100103
[15] J. Christensen, M. Kadic, M. Wegener, and O. Kraft, “Vibrant times for mechanical metamaterials,” MRS Communications, vol. 5, no. 3, pp. 453–462, 2015. https://doi.org/10.1557/mrc.2015.51
[16] U. M. Dilberoglu, B. Gharehpapagh, U. Yaman, and M. Dolen, “The role of additive manufacturing in the era of Industry 4.0,” Procedia Manufacturing, vol. 11, pp. 545–554, 2017. https://doi.org/10.1016/j.promfg.2017.07.148
[17] Q. Yan, H. Dong, J. Su, J. Han, B. Song, Q. Wei, and Y. Shi, “A review of 3D printing technology for medical applications,” Engineering, vol. 4, no. 5, pp. 729–742, 2018. https://doi.org/10.1016/j.eng.2018.07.021
[18] X. Yu, J. Zhou, H. Liang, Z. Jiang, and L. Wu, “Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review,” Progress in Materials Science, vol. 94, pp. 114–173, 2018. https://doi.org/10.1016/j.pmatsci.2017.12.003
[19] M. Srivastava, S. Rathee, V. Patel, A. Kumar, and P. G. Koppad, “A review of various materials for additive manufacturing: Recent trends and processing issues,” Journal of Materials Research and Technology, vol. 21, pp. 2612–2641, 2022. https://doi.org/10.1016/j.jmrt.2022.10.015
[20] B. Song, S. Zhang, L. Zhang, and Y. Shi, “Development trends and challenges of additive manufacturing metamaterials,” Engineering, vol. 6, Art. no. S2095809924006659, 2024. https://doi.org/10.1016/j.eng.2024.11.014
[21] A. Shah, “Emerging trends in robotic aided additive manufacturing,” Materials Today: Proceedings, vol. 62, pp. 7231–7237, 2022. https://doi.org/10.1016/j.matpr.2022.03.680
[22] J. Ribeiro, M. Rodríguez-Martín, J. Barreiro, A. Fernández-Abia, R. García-Martín, J. Rocha, and S. Martínez-Pellitero, “New trends of additive manufacturing using materials based on natural fibers and minerals: A systematic review,” Heliyon, vol. 11, no. 2, Art. no. e41993, 2025. https://doi.org/10.1016/j.heliyon.2025.e41993
[23] B. Lu, D. Li, and X. Tian, “Development trends in additive manufacturing and 3D printing,” Engineering, vol. 1, no. 1, pp. 85–89, 2015. https://doi.org/10.15302/J-ENG-2015012
[24] N. Khan and A. Riccio, “A systematic review of design for additive manufacturing of aerospace lattice structures: Current trends and future directions,” Progress in Aerospace Sciences, vol. 149, Art. no. 101021, 2024. https://doi.org/10.1016/j.paerosci.2024.101021
[25] J. Fan, L. Zhang, S. Wei, Z. Zhang, S.-K. Choi, B. Song, and Y. Shi, “A review of additive manufacturing of metamaterials and developing trends,” Materials Today, vol. 50, pp. 303–328, 2021. https://doi.org/10.1016/j.mattod.2021.04.019
[26] P. B. A., L. N., A. Buradi, S. N., P. B. L., and V. R., “A comprehensive review of emerging additive manufacturing (3D printing technology): Methods, materials, applications, challenges, trends and future potential,” Materials Today: Proceedings, vol. 52, pp. 1309–1313, 2022. https://doi.org/10.1016/j.matpr.2021.11.059
[27] J. L. Amaya-Rivas, B. S. Perero, C. G. Helguero, J. L. Hurel, J. M. Peralta, F. A. Flores, and J. D. Alvarado, “Future trends of additive manufacturing in medical applications: An overview,” Heliyon, vol. 10, no. 5, Art. no. e26641, 2024. https://doi.org/10.1016/j.heliyon.2024.e26641
[28] G. Rasiya, A. Shukla, and K. Saran, “Additive manufacturing - a review,” Materials Today: Proceedings, vol. 47, pp. 6896–6901, 2021. https://doi.org/10.1016/j.matpr.2021.05.181
[29] L. Zhang, B. Song, A. Zhao, R. Liu, L. Yang, and Y. Shi, “Study on mechanical properties of honeycomb pentamode structures fabricated by laser additive manufacturing: Numerical simulation and experimental verification,” Composite Structures, vol. 226, Art. no. 111199, 2019. https://doi.org/10.1016/j.compstruct.2019.111199
[30] S. Jiang, D. Guo, L. Zhang, K. Li, B. Song, and Y. Huang, “Electropolishing-enhanced, high-precision 3D printing of metallic pentamode metamaterials,” Materials & Design, vol. 223, Art. no. 111211, 2022. https://doi.org/10.1016/j.matdes.2022.111211
[31] S. Wu, Z. Luo, Z. Li, S. Liu, and L.-C. Zhang, “Topological design of pentamode metamaterials with additive manufacturing,” Computer Methods in Applied Mechanics and Engineering, vol. 377, Art. no. 113708, 2021. https://doi.org/10.1016/j.cma.2021.113708
[32] R. Hedayati, S. Jedari Salami, Y. Li, M. Sadighi, and A. A. Zadpoor, “Semianalytical geometry-property relationships for some generalized classes of pentamodelike additively manufactured mechanical metamaterials,” Physical Review Applied, vol. 11, no. 3, Art. no. 034057, 2019. https://doi.org/10.1103/PhysRevApplied.11.034057
[33] K. Mohammadi, M. R. Movahhedy, I. Shishkovsky, and R. Hedayati, “Hybrid anisotropic pentamode mechanical metamaterial produced by additive manufacturing technique,” Applied Physics Letters, vol. 117, no. 6, Art. no. 061901, 2020. https://doi.org/10.1063/5.0014167
[34] L. Hu, K. Shi, X. Luo, J. Yu, B. Ai, and C. Liu, “Application of additively manufactured pentamode metamaterials in sodium/Inconel 718 heat pipes,” Materials, vol. 14, no. 11, Art. no. 3016, 2021. https://doi.org/10.3390/ma14113016
[35] Z. Zhang and B. Han, “Regulation mechanism of the shock waves in a pentamode lattice-ring structure subjected to collision,” Crystals, vol. 12, no. 12, Art. no. 1749, 2022. https://doi.org/10.3390/cryst12121749
[36] R. Hedayati, K. Mohammadi, S. J. Salami, N. Roudbarian, P. Nayyeri, M. M. Rafiee, and H. Bougherara, “Closed-form analytical relationships for pentamode metamaterials,” Composite Structures, vol. 344, Art. no. 118334, 2024. https://doi.org/10.1016/j.compstruct.2024.118334
[37] K. Mohammadi, M. Shafia, J. Akbari, and R. Hedayati, “Pentamodes: Effect of unit cell topology on mechanical properties,” Results in Engineering, vol. 22, Art. no. 101982, 2024. https://doi.org/10.1016/j.rineng.2024.101982
[38] P. N. Lymperopoulos, E. E. Theotokoglou, D. Dragatogiannis, D. Karalekas, and C. Matsika-Klossa, “Additive manufacturing, numerical and experimental analyses for pentamode metamaterials,” Infrastructures, vol. 9, no. 10, Art. no. 172, 2024. https://doi.org/10.3390/infrastructures9100172
[39] D. Guo, S. Jiang, Y. Zhou, L. Zhang, K. Li, B. Song, and Y. Huang, “Ultrahigh compression-shear ratio of sandwich pentamode metamaterials,” Composite Structures, vol. 322, Art. no. 117331, 2023. https://doi.org/10.1016/j.compstruct.2023.117331
[40] Z. Wei, Z. Hu, R. Zhu, Y. Chen, and G. Hu, “A transformable anisotropic 3D pentamode metamaterial,” Materials & Design, vol. 234, Art. no. 112306, 2023. https://doi.org/10.1016/j.matdes.2023.112306
[41] C. W. Cushing, M. J. Kelsten, X. Su, P. S. Wilson, M. R. Haberman, and A. N. Norris, “Design and characterization of a three-dimensional anisotropic additively manufactured pentamode material,” The Journal of the Acoustical Society of America, vol. 151, no. 1, pp. 168–179, 2022. https://doi.org/10.1121/10.0009161
[42] A. Amendola, C. J. Smith, R. Goodall, F. Auricchio, L. Feo, G. Benzoni, and F. Fraternali, “Experimental response of additively manufactured metallic pentamode materials confined between stiffening plates,” Composite Structures, vol. 142, pp. 254–262, 2016. https://doi.org/10.1016/j.compstruct.2016.01.091
[43] L. Zhang, B. Song, R. Liu, A. Zhao, J. Zhang, L. Zhuo, G. Tang, and Y. Shi, “Effects of structural parameters on the Poisson’s ratio and compressive modulus of 2D pentamode structures fabricated by selective laser melting,” Engineering, vol. 6, no. 1, pp. 56–67, 2020. https://doi.org/10.1016/j.eng.2019.06.009
[44] X. Chen, L. Cai, and J. Wen, “Broadband transformation acoustic waveguide with anisotropic density based on pentamode metamaterials,” Frontiers in Materials, vol. 9, Art. no. 860126, 2022. https://doi.org/10.3389/fmats.2022.860126
[45] Y. Huang and X. Zhang, “Pentamode metamaterials with ultra-low-frequency single-mode band gap based on constituent materials,” Journal of Physics D: Applied Physics, vol. 33, no. 18, Art. no. 185301, 2021. https://doi.org/10.1088/1361-6463/ab1234
[46] A. Zhao, C. Liu, H. Zou, H. Jia, M. Zhang, T. Wu, H. Chen, X. Zhang, and Z. Wang, “Massive and fast fabrication of pentamode devices through a multiphase honeycomb-corrugation configuration,” Materials & Design, vol. 228, Art. no. 111816, 2023. https://doi.org/10.1016/j.matdes.2023.111816
[47] D. E. Quadrelli, M. A. Casieri, G. Cazzulani, S. La Riviera, and F. Braghin, “Experimental validation of a broadband pentamode elliptical-shaped cloak for underwater acoustics,” Extreme Mechanics Letters, vol. 49, Art. no. 101526, 2021. https://doi.org/10.1016/j.eml.2021.101526
[48] A. Amendola, G. Benzoni, and F. Fraternali, “Non-linear elastic response of layered structures, alternating pentamode lattices and confinement plates,” Composites Part B: Engineering, vol. 115, pp. 117–123, 2017. https://doi.org/10.1016/j.compositesb.2016.10.027
[49] C. Cai, C. Han, J. Wu, Z. Wang, and Q. Zhang, “Tuning method of phononic band gaps of locally resonant pentamode metamaterials,” Journal of Physics D: Applied Physics, vol. 52, no. 4, Art. no. 045601, 2019. https://doi.org/10.1088/1361-6463/aaf123
[50] Q. Li, K. Wu, and M. Zhang, “Two-dimensional composite acoustic metamaterials of rectangular unit cell from pentamode to band gap,” Crystals, vol. 11, no. 12, Art. no. 1457, 2021. https://doi.org/10.3390/cryst11121457
[51] Y. Huang, X. Zhang, H. Li, and C. Cai, “Core-shell pentamode metamaterials with broader mechanical response and higher sensitivity,” Physica Scripta, vol. 96, no. 5, Art. no. 055802, 2021. https://doi.org/10.1088/1402-4896/abf234
[52] P. N. Lymperopoulos and E. E. Theotokoglou, “Numerical analyses of pentamodes metamaterials behavior under harmonic loading conditions,” European Journal of Mechanics - A/Solids, vol. 111, Art. no. 105536, 2025. https://doi.org/10.1016/j.euromechsol.2024.105536
[53] Y. Chen, X. Liu, and G. Hu, “Latticed pentamode acoustic cloak,” Scientific Reports, vol. 5, no. 1, Art. no. 15745, 2015. https://doi.org/10.1038/srep15745
[54] G. Li, Y. Chen, and H. He, “Design method and infeasibility criteria for honeycomb pentamode cloak,” Physics Letters A, vol. 442, Art. no. 128174, 2022. https://doi.org/10.1016/j.physleta.2022.128174
[55] Y. Chen, X. Liu, and G. Hu, “Influences of imperfectness and inner constraints on an acoustic cloak with unideal pentamode materials,” Journal of Sound and Vibration, vol. 458, pp. 62–73, 2019. https://doi.org/10.1016/j.jsv.2019.06.005
[56] X. Chen, N. Laforge, Q. Ji, H. Tan, J. Liang, G. Ulliac, J. Moughames, S. Adrar, V. Laude, and M. Kadic, “Introduction to mechanical metamaterials and their effective properties,” Comptes Rendus Physique, vol. 21, no. 7–8, pp. 751–765, 2021. https://doi.org/10.5802/crphys.30
[57] Y. An, H. Zou, and A. Zhao, “Broadband waterborne multiphase pentamode metastructure with simultaneous wavefront manipulation and energy absorption capabilities,” Materials, vol. 16, no. 14, Art. no. 5051, 2023. https://doi.org/10.3390/ma16145051
[58] H.-W. Dong, S.-D. Zhao, X.-B. Miao, C. Shen, X. Zhang, Z. Zhao, C. Zhang, Y.-S. Wang, and L. Cheng, “Customized broadband pentamode metamaterials by topology optimization,” Journal of the Mechanics and Physics of Solids, vol. 152, Art. no. 104407, 2021. https://doi.org/10.1016/j.jmps.2021.104407
[59] Y. Ge, X. Liu, and G. Hu, “Design of elliptical underwater acoustic cloak with truss-latticed pentamode materials,” Theoretical and Applied Mechanics Letters, vol. 12, no. 4, Art. no. 100346, 2022. https://doi.org/10.1016/j.taml.2022.100346
[60] L. He, L. Cai, and X. Chen, “Pentamode-based coding metasurface for underwater acoustic stealth,” Journal of Applied Mathematics and Physics, vol. 9, no. 7, pp. 1829–1836, 2021. https://doi.org/10.4236/jamp.2021.97117
[61] G. Li, L. Tang, V. Sorokin, and S. Wang, “CVAE-based inverse design of two-dimensional honeycomb pentamode metastructure for acoustic cloaking,” Thin-Walled Structures, vol. 206, Art. no. 112623, 2025. https://doi.org/10.1016/j.tws.2024.112623
[62] Z. Luo, Q. Zhou, and P. Guo, “Microstructure design and performance optimization of constant modulus pentamode materials acoustic cloak,” Crystals, vol. 12, no. 11, Art. no. 1572, 2022. https://doi.org/10.3390/cryst12111572
[63] Z. Luo, Q. Zhou, and P. Guo, “Optimization design method of pressure pentamode materials,” AIP Advances, vol. 12, no. 11, Art. no. 115010, 2022. https://doi.org/10.1063/5.0120592
[64] X. Nie, Y. Chen, and X. Liu, “Scattering analysis and optimization of spherical acoustic cloak with unideal pentamode material,” Acta Mechanica Solida Sinica, vol. 33, no. 3, pp. 347–360, 2020. https://doi.org/10.1007/s10338-019-00139-x
[65] D. E. Quadrelli, G. Cazzulani, S. La Riviera, and F. Braghin, “Acoustic scattering reduction of elliptical targets via pentamode near-cloaking based on transformation acoustics in elliptic coordinates,” Journal of Sound and Vibration, vol. 512, Art. no. 116396, 2021. https://doi.org/10.1016/j.jsv.2021.116396
[66] Z. Sun, H. Jia, H. Zhao, S. Wang, H. Zhang, and J. Yang, “Underwater acoustic bottle beam generated by latticed pentamode metasurface,” Journal of Applied Physics, vol. 137, no. 2, Art. no. 020001, 2022.
[67] Z. Sun, L. Zhao, and J. Yang, “Underwater acoustic self-focusing and bending in conformal Mikaelian lens by pentamode metafluid,” Journal of Physics D: Applied Physics, vol. 56, no. 50, Art. no. 505304, 2023.
[68] M. Zaremanesh and A. Bahrami, “Two-dimensional honeycomb lattice structure for underwater acoustic cloaking using pentamode materials,” Physica Scripta, vol. 99, no. 1, Art. no. 015946, 2024.
[69] B. Zhao, P. Wang, D. Wang, and G. Hu, “Underwater directional acoustic source based on pentamode material,” Acta Mechanica Solida Sinica, vol. 37, no. 1, pp. 1–9, 2024. https://doi.org/10.1007/s10338-023-00442-8
[70] S.-D. Zhao, H.-W. Dong, K. Wang, Y.-S. Wang, and C. Zhang, “Topology-optimized reflection-type pentamode metasurfaces for broadband underwater beam engineering,” SSRN Electronic Journal, vol. 135, Art. no. 10351, 2022. https://doi.org/10.2139/ssrn.4300045
[71] Y. Chen, X. Liu, and G. Hu, “Latticed pentamode acoustic cloak,” Scientific Reports, vol. 5, no. 1, Art. no. 15745, 2015. https://doi.org/10.1038/srep15745
[72] Y. Chu, Y. Li, C. Cai, G. Liu, Z. Wang, and Z. Xu, “Ultrawide bandgap pentamode metamaterials with an asymmetric double-cone outside profile,” Journal of Physics D: Applied Physics, vol. 51, no. 12, Art. no. 125103, 2018.
[73] Q. Li and M. Zhang, “Composite hexagonal pentamode acoustic metamaterials with tailored properties,” Journal of Physics D: Applied Physics, vol. 32, no. 47, Art. no. 475701, 2020.
[74] Y. Chu, Z. Wang, and Z. Xu, “Broadband high-efficiency controllable asymmetric propagation by pentamode acoustic metasurface,” Physics Letters A, vol. 384, no. 11, Art. no. 126230, 2020. https://doi.org/10.1016/j.physleta.2019.126230
[75] Z. Li, Z. Luo, L.-C. Zhang, and C.-H. Wang, “Topological design of pentamode lattice metamaterials using a ground structure method,” Materials & Design, vol. 202, Art. no. 109523, 2021. https://doi.org/10.1016/j.matdes.2021.109523
[76] A. Zhao, C. Liu, H. Zou, H. Jia, M. Zhang, T. Wu, H. Chen, X. Zhang, and Z. Wang, “Massive and fast fabrication of pentamode devices through a honeycomb-corrugation configuration,” SSRN Electronic Journal, vol. 228, Art. no. 111816, 2022. https://doi.org/10.2139/ssrn.4184525
[77] Q. Li and M. Zhang, “Elastic metamaterials of hexagonal unit cells with double-cone arms from pentamode to band gap at low frequencies,” Crystals, vol. 12, no. 5, Art. no. 604, 2022. https://doi.org/10.3390/cryst12050604
[78] C. Cai, X. Wang, Q. Wang, M. Li, G. He, Z. Wang, and Y. Qin, “Design and optimization of three-dimensional composite multilayer cylindrical pentamode metamaterials for controlling low frequency acoustic waves,” Scientific Reports, vol. 12, no. 1, Art. no. 5594, 2022. https://doi.org/10.1038/s41598-022-09313-7
[79] B. Kumar, A. Banerjee, and B. Manna, “Effect of finite mass on phononic band structure of face-centered pentamodal lattice,” Mechanics Research Communications, vol. 124, Art. no. 103933, 2022. https://doi.org/10.1016/j.mechrescom.2022.103933
[80] Z. Zou, F. Xu, Y. Pan, X. Niu, T. Fang, and C. Zeng, “Parametric analysis and multi-objective optimization of pentamode metamaterial,” Sustainability, vol. 15, no. 4, Art. no. 3421, 2023. https://doi.org/10.3390/su15043421
[81] Y. Chu, T. Sun, Z. Wang, Z. Zhang, and M. Chen, “Low-frequency broadband acoustic modulation mechanism of composite pentamode metamaterials,” Physics Letters A, vol. 491, Art. no. 129212, 2023. https://doi.org/10.1016/j.physleta.2023.129212
[82] Z. Zou, F. Xu, Y. Pan, and T. Fang, “Bandgap properties and multi-objective optimization of double-cone pentamode metamaterials with curved side,” Physica Scripta, vol. 98, no. 3, Art. no. 035833, 2023.
[83] Y. Chu, Y. Hu, G. Wang, and Z. Wang, “Dual-functional pentamode metamaterial with water-like and topological transmission properties,” Journal of Physics D: Applied Physics, vol. 36, no. 32, Art. no. 325701, 2024.
[84] J. Chen, J. Liu, and X. Liu, “Broadband underwater acoustic carpet cloak based on pentamode materials under normal incidence,” AIP Advances, vol. 8, no. 8, Art. no. 085024, 2018. https://doi.org/10.1063/1.5042451
[85] Y. Liu, Y.-F. Li, and X.-Z. Liu, “Manipulation of acoustic wavefront by transmissive metasurface based on pentamode metamaterials,” Journal of Physics D: Applied Physics, vol. 28, no. 2, Art. no. 024301, 2019.
[86] Z. Sun, Y. Shi, X. Sun, H. Jia, Z. Jin, K. Deng, and J. Yang, “Underwater acoustic multiplexing communication by pentamode metasurface,” Journal of Physics D: Applied Physics, vol. 54, no. 20, Art. no. 205303, 2021.
[87] P. N. Lymperopoulos and E. E. Theotokoglou, “Computational analysis of pentamode metamaterials for antiseismic design,” Procedia Structural Integrity, vol. 26, pp. 263–268, 2020. https://doi.org/10.1016/j.prostr.2020.06.033
[88] C. Cai, Z. Wang, Q. Li, Z. Xu, and X. Tian, “Pentamode metamaterials with asymmetric double-cone elements,” Journal of Physics D: Applied Physics, vol. 48, no. 17, Art. no. 175103, 2015.
[89] F. Fabbrocino and A. Amendola, “Discrete-to-continuum approaches to the mechanics of pentamode bearings,” Composite Structures, vol. 167, pp. 219–226, 2017. https://doi.org/10.1016/j.compstruct.2017.01.073
[90] V. A. Skripnyak, M. Chirkov, E. G. Skripnyak, and V. V. Skripnyak, “Pentamode metamaterials under dynamic loading,” in Proc. 7th Int. Congr. on Energy Fluxes and Radiation Effects (EFRE), pp. 1101–1105, 2020. https://doi.org/10.1109/EFRE47760.2020.9242159
[91] O. E. Sapountzaki, A. E. Kampitsis, and N. D. Lagaros, “Failure mechanisms of anisotropic pentamode-based bridge bearings: A dynamic analysis,” Engineering Structures, vol. 301, Art. no. 117292, 2024. https://doi.org/10.1016/j.engstruct.2023.117292
[92] G. W. Milton and M. Camar-Eddine, “Near optimal pentamodes as a tool for guiding stress while minimizing compliance in 3D-printed materials: A complete solution to the weak G-closure problem for 3D-printed materials,” Journal of the Mechanics and Physics of Solids, vol. 114, pp. 194–208, 2018. https://doi.org/10.1016/j.jmps.2018.02.003
[93] S. Bronder and A. Jung, “Pentamode structures optimized by machine learning with adaptive sampling,” Advanced Engineering Materials, vol. 26, no. 24, Art. no. 2302073, 2024. https://doi.org/10.1002/adem.202302073
[94] G. Brambilla, S. Cominelli, M. Verbicaro, G. Cazzulani, and F. Braghin, “High bulk modulus pentamodes: The three-dimensional metal water,” Extreme Mechanics Letters, vol. 74, Art. no. 102267, 2025. https://doi.org/10.1016/j.eml.2024.102267
[95] A. Amendola, M. Titirla, R. Penna, and F. Fabbrocino, “On the design of performance-based pentamode bearings,” in Proc. 6th Int. Conf. on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN), 2015.
[96] A. Amendola, I. Mascolo, and G. Benzoni, “On the mechanical response of multilayered pentamode lattices equipped with hinged and rigid nodes,” PSU Research Review, vol. 2, no. 2, pp. 138–144, 2018. https://doi.org/10.1108/PRR-07-2018-0022
[97] Y. Huang, X. Zhang, M. Kadic, and G. Liang, “Stiffer, stronger and centrosymmetrical class of pentamodal mechanical metamaterials,” Materials, vol. 12, no. 21, Art. no. 3470, 2019. https://doi.org/10.3390/ma12213470
[98] P. N. Lymperopoulos and E. E. Theotokoglou, “Numerical investigation of pentamodes crack propagation,” Procedia Structural Integrity, vol. 33, pp. 818–823, 2021. https://doi.org/10.1016/j.prostr.2021.10.091
[99] S. E. Rodriguez, E. P. Calius, A. Khatibi, A. C. Orifici, and R. Das, “An automated design framework for composite mechanical metamaterials and its application to 2D pentamode materials,” International Journal of Mechanical Sciences, vol. 276, Art. no. 109393, 2024. https://doi.org/10.1016/j.ijmecsci.2024.109393
[100] P. N. Lymperopoulos and E. E. Theotokoglou, “Dynamic computational analyses of pentamodes structures,” Procedia Structural Integrity, vol. 47, pp. 274–281, 2023. https://doi.org/10.1016/j.prostr.2023.07.098
[101] P. N. Lymperopoulos and E. E. Theotokoglou, “Numerical investigation of pentamode mechanical metamaterials,” WSEAS Transactions on Applied and Theoretical Mechanics, vol. 17, pp. 47–55, 2022.
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