A Comprehensive Parametric Analysis of Geometric Effects on the Natural Frequencies of Auxetic and Honeycomb Beams

[1]          C. Pan, Y. Han, J. Lu, Design and optimization of lattice structures: A review, Applied Sciences, Vol. 10, No. 18, pp. 6374, 2020.

[2]          X. Ren, R. Das, P. Tran, T. D. Ngo, Y. M. Xie, Auxetic metamaterials and structures: A review, Smart Materials and Structures, 2018.

[3]          M. S. Al-Khazraji, Review on impact, crushing response and applications of re-entrant core sandwich structures, Aircraft Engineering and Aerospace Technology, 2024.

[4]          F. Baertsch, A. Ameli, T. Mayer, Finite-Element Modeling and Optimization of 3D-Printed Auxetic Reentrant Structures with Stiffness Gradient under Low-Velocity Impact, Journal of Engineering Mechanics, Vol. 147, No. 7, 2021.

[5]          S. K. Bhullar, D. Rana, H. Lekesiz, A. C. Bedeloglu, J. Ko, Y. Cho, Z. Aytac, T. Uyar, M. Jun, M. Ramalingam, Design and fabrication of auxetic PCL nanofiber membranes for biomedical applications, Materials Science and Engineering C, Vol. 81, 2017.

[6]          Q. Hu, G. Lu, K. M. Tse, Compressive and tensile behaviours of 3D hybrid auxetic-honeycomb lattice structures, International Journal of Mechanical Sciences, Vol. 263, 2024.

[7]          Z. Wang, C. Luan, G. Liao, J. Liu, X. Yao, J. Fu, Progress in auxetic mechanical metamaterials: structures, characteristics, manufacturing methods, and applications, Advanced Engineering Materials, Vol. 22, No. 10, pp. 2000312, 2020.

[8]          D. V. Truong, H. Nguyễn, R. Fangueiro, F. Ferreira, Q. Nguyễn, Auxetic materials and structures in the automotive industry: Applications and insights, Journal of Reinforced Plastics and Composites, pp. 07316844251334174, 2025.

[9]          H. Mallek, H. Mellouli, M. Allouch, M. Wali, F. Dammak, Energy absorption of 3D-printed PETG and PETG/CF sandwich structures with cellular cores subjected to low-velocity impact: Experimental and numerical analysis, Engineering Structures, Vol. 327, pp. 119653-119653, 2025.

[10]        Y. Zhang, W. Z. Jiang, W. Jiang, X. Y. Zhang, J. Dong, Y. M. Xie, K. E. Evans, X. Ren, Recent Advances of Auxetic Metamaterials in Smart Materials and Structural Systems, Advanced Functional Materials, pp. 2421746, 2025.

[11]        M. Hedayatian, A. R. Daneshmehr, G. H. Liaghat, The Efficiency of Auxetic Cores in Sandwich Beams Subjected to Low-Velocity Impact, International Journal of Applied Mechanics, Vol. 12, No. 6, 2020.

[12]        V. Siniauskaya, H. Wang, Y. Liu, Y. Chen, M. Zhuravkov, Y. Lyu, A review on the auxetic mechanical metamaterials and their applications in the field of applied engineering, Frontiers in Materials, Vol. 11, pp. 1453905, 2024.

[13]        S. M. R. Khalili, S. M. A. Alavi, Computation of the homogenized linear elastic response of 2D microcellular re-entrant auxetic honeycombs based on modified strain gradient theory, Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 45, No. 1, pp. 19-19, 2022.

[14]        K. Wei, Y. Peng, Z. Qu, Y. Pei, D. Fang, A cellular metastructure incorporating coupled negative thermal expansion and negative Poisson's ratio, International Journal of Solids and Structures, Vol. 150, 2018.

[15]        A. R. Damanpack, M. Bodaghi, W. H. Liao, Experimentally validated multi-scale modeling of 3D printed hyper-elastic lattices, International Journal of Non-Linear Mechanics, Vol. 108, 2019.

[16]        J. Ma, H. Zhang, T.-U. Lee, H. Lu, Y. M. Xie, N. S. Ha, Auxetic behavior and energy absorption characteristics of a lattice structure inspired by deep-sea sponge, Composite Structures, Vol. 354, pp. 118835-118835, 2025.

[17]        M. Hosseini, H. Mazaheri, Mechanical behavior of graded combined auxetic-honeycomb structures, International Journal of Mechanical Sciences, Vol. 276, pp. 109223-109223, 2024.

[18]        B. M. M, R. M, C. Raja S, P. T. Doutre, F. Vignat, Geometric configuration and parametric evaluation of auxetic meta-materials for enhanced plastic energy dissipation in blast scenarios, Journal of Computational Applied Mechanics, Vol. 56, No. 1, pp. 127-144, 2025.

[19]        M. Bodaghi, A. Serjouei, A. Zolfagharian, M. Fotouhi, H. Rahman, D. Durand, Reversible energy absorbing meta-sandwiches by FDM 4D printing, International Journal of Mechanical Sciences, Vol. 173, 2020.

[20]        S. Rezaei, J. Kadkhodapour, R. Hamzehei, B. Taherkhani, A. P. Anaraki, S. Dariushi, Design and modeling of the 2D auxetic metamaterials with hyperelastic properties using topology optimization approach, Photonics and Nanostructures - Fundamentals and Applications, Vol. 43, 2021.

[21]        J. Lotfi, S. M. R. Khalili, A. R. Damanpack, Micro-Macro Analysis of Hyperelastic Auxetic Lattice Structures under Finite-Strain Regime, International Journal of Mechanical Sciences, pp. 109246-109246, 2024/3//, 2024.

[22]        K. M. Bora, S. K. Varshney, C. S. Kumar, Non local analytical and numerical modelling of re-entrant auxetic honeycomb, Engineering Research Express, Vol. 6, No. 2, pp. 025533-025533, 2024.

[23]        M. Mohammadimehr, S. V. Okhravi, S. M. Akhavan Alavi, Free vibration analysis of magneto-electro-elastic cylindrical composite panel reinforced by various distributions of CNTs with considering open and closed circuits boundary conditions based on FSDT, JVC/Journal of Vibration and Control, Vol. 24, No. 8, 2018.

[24]        M. Jafari, M. Mohammadimehr, Forced vibration control of Timoshenko’s micro sandwich beam with CNT/GPL/CNR reinforced composites integrated by piezoelectric on Kerr’s elastic foundation using MCST, Journal of Computational Applied Mechanics, Vol. 56, No. 1, pp. 15-42, 2025.

[25]        G. Şakar, F. Ç. Bolat, The Free Vibration Analysis of Honeycomb Sandwich Beam Using 3D and Continuum Model, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, Vol. 9, No. 6, 2015.

[26]        X. Chen, Z. Feng, Dynamic behaviour of a thin laminated plate embedded with auxetic layers subject to in-plane excitation, Mechanics Research Communications, Vol. 85, 2017.

[27]        M. H. Zamani, M. Heidari-Rarani, K. Torabi, A novel graded auxetic honeycomb core model for sandwich structures with increasing natural frequencies, Journal of Sandwich Structures and Materials, Vol. 24, No. 2, 2022.

[28]        Y. S. Kushwaha, N. S. Hemanth, N. D. Badgayan, S. K. Sahu, Free vibration analysis of PLA based auxetic metamaterial structural composite using finite element analysis, Materials Today: Proceedings, Vol. 56, 2022.

[29]        W. Jiang, J. Zhou, J. Liu, M. Zhang, W. Huang, Free vibration behaviours of composite sandwich plates with reentrant honeycomb cores, Applied Mathematical Modelling, Vol. 116, 2023.

[30]        R. Hosseini, M. Babaei, A. Naddaf, The influences of various auxetic cores on natural frequencies and forced vibration behavior of sandwich beam fabricated by 3D printer based on third -order shear deformation theory, Journal of Computational Applied Mechanics, Vol. 54, No. 2, 2023.

[31]        T. Q. Quan, V. M. Anh, N. D. Duc, Natural frequency analysis of sandwich plate with auxetic honeycomb core and CNTRC face sheets using analytical approach and artificial neural network, Aerospace Science and Technology, Vol. 144, 2024.

[32]        N. Namvar, A. Zolfagharian, F. Vakili-Tahami, M. Bodaghi, Reversible energy absorption of elasto-plastic auxetic, hexagonal, and AuxHex structures fabricated by FDM 4D printing, Smart Materials and Structures, Vol. 31, No. 5, 2022/5//, 2022.

[33]        A. M. M. Nazmul Ahsan, B. Khoda, Characterizing Novel Honeycomb Infill Pattern for Additive Manufacturing, Journal of Manufacturing Science and Engineering, Transactions of the ASME, Vol. 143, No. 2, 2021.

[34]        S. M. R. Khalili, A. R. Damanpack, N. Nemati, K. Malekzadeh, Free vibration analysis of sandwich beam carrying sprung masses, International Journal of Mechanical Sciences, Vol. 52, No. 12, 2010.

[35]        A. M. Ahmed, A. M. Rifai, Euler-Bernoulli and Timoshenko Beam Theories Analytical and Numerical Comprehensive Revision, European Journal of Engineering and Technology Research, Vol. 6, No. 7, 2021.

[36]        J. N. Reddy, 2015, An Introduction to Nonlinear Finite Element Analysis, 2nd Edn,

[37]        M. J. Jweeg, M. Al-Waily, K. K. Resan, Introduction to finite element method: bar and beam applications,  in: Energy Methods and Finite Element Techniques, Eds., 2022.

[38]        B. Wang, J. Bai, S. Lu, W. Zuo, An open source MATLAB solver for contact finite element analysis, Advances in Engineering Software, Vol. 199, pp. 103798-103798, 2025.

[39]        T. Wang, J. Xu, Z. Qi, T. Zhao, Shear stress correction in Euler-Bernoulli beam theory, Journal of Mechanical Science and Technology, 2025.

[40]        A. R. Damanpack, M. Bodaghi, W. H. Liao, Contact/impact modeling and analysis of 4D printed shape memory polymer beams, Smart Materials and Structures, Vol. 29, No. 8, 2020.

[41]        S. S. Rao, Mechanical Vibrations Sixth Edition in SI Units, Mechanical Vibrations, 2018.

[42]        M. Wallbanks, M. F. Khan, M. Bodaghi, A. Triantaphyllou, A. Serjouei, On the design workflow of auxetic metamaterials for structural applications, Smart Materials and Structures, 2022.

[43]        F. Xia, Y. Durandet, P. J. Tan, D. Ruan, Three-point bending performance of sandwich panels with various types of cores, Thin-Walled Structures, Vol. 179, pp. 109723-109723, 2022/10//, 2022.

[44]        J. Zhang, X. Zhu, X. Yang, W. Zhang, Transient nonlinear responses of an auxetic honeycomb sandwich plate under impact loads, International Journal of Impact Engineering, Vol. 134, 2019.

[45]        S. C. Chapra, R. P. Canale, 2021, Numerical Methods for Engineers, Eighth Edition,

[46]        A. Daman Pak, J. Lotfi, S. M. R. Khalili, A Finite-Strain Simulation of 3D Printed Airless Tires, Journal of Engineering Mechanics, Vol. 149, No. 10, 2023/10//, 2023.