Abstract
The primary objective of the proposed work was to present a theoretical and computational procedure to predict the effective elastic properties of a periodic honeycomb structure. In the theoretical framework, the effective orthotropic elastic properties were determined using the strain-energy approach. Whereas in the computational procedure, a homogenization technique based on the finite element (FE) method in conjunction with periodic boundary conditions (PBCs) was used to determine the equivalent properties of the honeycomb core. A suitable representative cell element (RCE) was chosen for this purpose. The computed effective elastic properties were compared with those obtained from the strain-energy approach, and the reference results were in good agreement with each other. Subsequently, the obtained elastic properties were used as material parameters of the sandwich structure comprising homogenized core structure and the face sheets for the FE analysis of the 3-point bend test (3PBT) of a sandwich structure, edge-compression test, and buckling problems. The results were compared with those obtained from the direct FE simulation of the honeycomb core structure. The comparison showed that the results were satisfactory, with a significant reduction in the computational time. Finally, the modal analysis was performed to reaffirm the efficiency of the presented procedure.
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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
References
Abdelal, G.F., Atef, A.: Thermal fatigue analysis of solar panel structure for micro-satellite applications. Int. J. Mech. Mater. Des. 4, 53–62 (2008). https://doi.org/10.1007/s10999-008-9057-3
Balawi, S., Abot, J.L.: The effect of honeycomb relative density on its effective in-plane elastic moduli: an experimental study. Compos. Struct. 84, 293–299 (2008). https://doi.org/10.1016/j.compstruct.2007.08.009
Burton, W.S., Noor, A.K.: Assessment of continuum models for sandwich panel honeycomb cores. Comput. Methods Appl. Mech. Eng. 145, 341–360 (1997)
Catapano, A., Montemurro, M.: A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part II: the optimisation strategy. Compos. Struct. 118, 677–690 (2014a). https://doi.org/10.1016/j.compstruct.2014.07.058
Catapano, A., Montemurro, M.: A multi-scale approach for the optimum design of sandwich plates with honeycomb core. Part I: homogenisation of core properties. Compos. Struct. 118, 664–676 (2014b). https://doi.org/10.1016/j.compstruct.2014.07.057
Cesari, F., Furgiuele, F.M., Maletta, C.: The determination of stress distribution and elastic properties for heterogeneous materials with hybrid finite element. Int. J. Mech. Mater. Des. 2, 1–13 (2005). https://doi.org/10.1007/s10999-005-3309-2
Chen, D.H., Ozaki, S.: Analysis of in-plane elastic modulus for a hexagonal honeycomb core: effect of core height and proposed analytical method. Compos. Struct. 88, 17–25 (2009). https://doi.org/10.1016/j.compstruct.2008.02.021
Czekanski, A., Elbestawi, M.A., Meguid, S.A.: On the FE modeling of closed-cell aluminum foam. Int. J. Mech. Mater. Des. 2, 23–34 (2005). https://doi.org/10.1007/s10999-005-0518-7
Dai, G., Zhang, W.: Cell size effect analysis of the effective Young’s modulus of sandwich core. Comput. Mater. Sci. 46, 744–748 (2009). https://doi.org/10.1016/j.commatsci.2009.04.033
Dirrenberger, J., Forest, S., Jeulin, D.: Effective elastic properties of auxetic microstructures: anisotropy and structural applications. Int. J. Mech. Mater. Des. 9, 21–33 (2013). https://doi.org/10.1007/s10999-012-9192-8
Gao, Y., Huang, H.: Equivalent damper model for honeycomb structures. Int. J. Mech. Mater. Des. 18, 475–490 (2022). https://doi.org/10.1007/s10999-021-09578-2
Gibson, L.J., Ashby, M.F.: Cellular solids: structure and properties. Pergamon Press, Oxford (1988)
Gibson, L.J., Ashby, M.F.: Cellular Solids: Structure and Properties. Cambridge Solid State Science Series (1997)
Gonella, S., Ruzzene, M.: Homogenization and equivalent in-plane properties of two-dimensional periodic lattices. Int. J. Solids Struct. 45, 2897–2915 (2008). https://doi.org/10.1016/j.ijsolstr.2008.01.002
Gornet, L., Marguet, S., Marckmann, G.: Numerical modelling of Nomex®honeycomb cores: failure and effective elastic properties. In: III European conference on computational mechanics. p. 509 (2006)
Goswami, S.: On the prediction of effective material properties of cellular hexagonal honeycomb core. J. Reinf. Plast. Compos. 25, 393–405 (2006). https://doi.org/10.1177/0731684405060567
Grediac, M.: A finite element study of the transverse shear in honeycomb cores. Int. J. Solids Struct. 30, 1777–1788 (1993)
Hassani, B.: A direct method to derive the boundary conditions of the homogenization equation for symmetric cells. Commun. Numer. Methods Eng. 12, 185–196 (1996)
Hedayati, R., Sadighi, M., Aghdam, M.M., Zadpoor, A.A.: Mechanical properties of additively manufactured thick honeycombs. Materials (basel) 9, 613 (2016). https://doi.org/10.3390/ma9080613
Hohe, J., Becker, W.: A mechanical model for two-dimensional cellular sandwich cores with general geometry. Comput. Mater. Sci. 19, 108–115 (2000). https://doi.org/10.1016/s0927-0256(00)00145-2
Hohe, J., Becker, W.: A refined analysis of the effective elasticity tensor for general cellular sandwich cores. Int. J. Solids Struct. 38, 3689–3717 (2001)
Ijaz, H., Saleem, W., Zain-Ul-Abdein, M., Mabrouki, T., Rubaiee, S., Salmeen Bin Mahfouz, A.: Finite Element analysis of bend test of sandwich structures using strain energy based homogenization method. Adv. Mater. Sci. Eng. (2017). https://doi.org/10.1155/2017/8670207
Kalamkarov, A.L., Andrianov, I.V., Danishevs’kyy, V.V.: Asymptotic homogenization of composite materials and structures. Appl. Mech. Rev. 62, 1–20 (2009). https://doi.org/10.1115/1.3090830
Kalamkarov, A.L., Saha, G.C., Georgiades, A.V.: General micromechanical modeling of smart composite shells with application to smart honeycomb sandwich structures. Compos. Struct. 79, 18–33 (2007). https://doi.org/10.1016/j.compstruct.2005.11.026
Kanit, T., N’Guyen, F., Forest, S., Jeulin, D., Reed, M., Singleton, S.: Apparent and effective physical properties of heterogeneous materials: representativity of samples of two materials from food industry. Comput. Methods Appl. Mech. Eng. 195, 3960–3982 (2006). https://doi.org/10.1016/j.cma.2005.07.022
Kelsey, S., Gellatly, R.A., Clark, B.W.: The shear modulus of foil honeycomb cores: a theoretical and experimental investigation on cores used in sandwich construction. Aircr. Eng. 30, 294–302 (1958)
Kumar, A., Muthu, N., Narayanan, R.G.: Equivalent in-plane elastic properties of periodic re-entrant honeycombs—strain-energy approach and FE modelling. In: Nirsanametla, Y., M. Khan, S., and Chowdhury, S. (eds.), International Conference on Experimental and Computational Methods in Manufacturing (ICECMM 2021). p. paper_48. Department of Mechanical Engineering North Eastern Regional Institute of Science and Technology, Nirjuli – 791109, Arunachal Pradesh, India (2021)
Li, X., Liu, Q., Zhang, J.: A micro-macro homogenization approach for discrete particle assembly—cosserat continuum modeling of granular materials. Int. J. Solids Struct. 47, 291–303 (2010). https://doi.org/10.1016/j.ijsolstr.2009.09.033
Liu, P., Liu, Y., Zhang, X.: Simulation of hyper-velocity impact on double honeycomb sandwich panel and its staggered improvement with internal-structure model. Int. J. Mech. Mater. Des. 12, 241–254 (2016). https://doi.org/10.1007/s10999-015-9300-7
Liu, Y., Liu, W., Gao, W.: Out-of-plane shear property analysis of Nomex honeycomb sandwich structure. J. Reinf. Plast. Compos. 40, 165–175 (2021). https://doi.org/10.1177/0731684420943285
Malek, S., Gibson, L.: Effective elastic properties of periodic hexagonal honeycombs. Mech. Mater. 91, 226–240 (2015). https://doi.org/10.1016/j.mechmat.2015.07.008
Masters, I.G., Evans, K.E.: Models for the elastic deformation of honeycombs. Compos. Struct. 35, 403–422 (1996)
Meguid, S.A., Cheon, S.S., El-Abbasi, N.: FE modelling of deformation localization in metallic foams. Finite Elem. Anal. Des. 38, 631–643 (2002). https://doi.org/10.1016/S0168-874X(01)00096-8
Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput. Methods Appl. Mech. Eng. 157, 69–94 (1998)
Niu, J., Choo, H.L., Sun, W., Mok, S.H.: Numerical study on load-bearing capabilities of beam-like lattice structures with three different unit cells. Int. J. Mech. Mater. Des. 14, 443–460 (2018). https://doi.org/10.1007/s10999-017-9384-3
Pan, S.D., Wu, L.Z., Sun, Y.G., Zhou, Z.G., Qu, J.L.: Longitudinal shear strength and failure process of honeycomb cores. Compos. Struct. 72, 42–46 (2006). https://doi.org/10.1016/j.compstruct.2004.10.011
Penado, F.E.: Effective elastic properties of honeycomb core with fiber-reinforced composite cells. Open J. Compos. Mater. 03, 89–96 (2013). https://doi.org/10.4236/ojcm.2013.34009
Qiu, C., Guan, Z., Jiang, S., Li, Z.: A method of determining effective elastic properties of honeycomb cores based on equal strain energy. Chin. J. Aeronaut. 30, 766–779 (2017). https://doi.org/10.1016/j.cja.2017.02.016
Rakesh, P., More, A., Kumar, M., Muthu, N.: Probabilistic failure prediction in a double composite cantilever beam with single and double source uncertainty. Compos. Struct. 279, 114870 (2022). https://doi.org/10.1016/j.compstruct.2021.114870
Saha, G.C., Kalamkarov, A.L.: Micromechanical thermoelastic model for sandwich composite shells made of generally orthotropic materials. J. Sandw. Struct. Mater. 11, 27–56 (2009)
Shedbale, A.S., Singh, I.V., Mishra, B.K.: Heterogeneous and homogenized models for predicting the indentation response of particle reinforced metal matrix composites. Int. J. Mech. Mater. Des. 13, 531–552 (2017). https://doi.org/10.1007/s10999-016-9352-3
Shoja-Senobar, M., Etemadi, E., Lezgy-Nazargah, M.: An analytical investigation of elastic–plastic behaviors of 3D warp and woof auxetic structures. Int. J. Mech. Mater. Des. 17, 545–561 (2021). https://doi.org/10.1007/s10999-021-09546-w
Somireddy, M., Czekanski, A., Singh, C.V.: Development of constitutive material model of 3D printed structure via FDM. Mater. Today Commun. 15, 143–152 (2018). https://doi.org/10.1016/j.mtcomm.2018.03.004
Sorohan, S., Constantinescu, D.M., Sandu, M., Sandu, A.G.: On the homogenization of hexagonal honeycombs under axial and shear loading. Part I: analytical formulation for free skin effect. Mech. Mater. 119, 74–91 (2018). https://doi.org/10.1016/j.mechmat.2017.09.003
Sorohan, S., Sandu, M., Constantinescu, D.M., Sandu, A.G.: On the evaluation of mechanical properties of honeycombs by using finite element analyses. INCAS Bull. 7, 135 (2015)
Sun, C.T., Vaidya, R.S.: Prediction of composite properties from a representative volume element. Compos. Sci. Technol. 56, 171–179 (1996)
Terada, K., Hori, M., Kyoya, T., Kikuchi, N.: Simulation of the multi-scale convergence in computational homogenization approaches. Int. J. Solids Struct. 37, 2285–2311 (2000). https://doi.org/10.1016/S0020-7683(98)00341-2
Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for representative volume elements of composites and applications. Int. J. Solids Struct. 40, 1907–1921 (2003). https://doi.org/10.1016/S0020-7683(03)00024-6
Xia, Z.H., Zhang, Y.F., Ellyin, F.: On application of periodic boundary conditions in micro/meso multi-scale analyses of composites. Key Eng. Mater. 345–346, 983–986 (2007). https://doi.org/10.4028/www.scientific.net/KEM.345-346.983
Zhang, J., Ashby, M.F.: The out-of-plane properties of honeycombs. Int. J. Mech. Sci. 34, 475–489 (1992)
Zhang, L., Liu, B., Gu, Y., Xu, X.H.: Modelling and characterization of mechanical properties of optimized honeycomb structure. Int. J. Mech. Mater. Des. 16, 155–166 (2020). https://doi.org/10.1007/s10999-019-09462-0
Zhang, Q., Yang, X., Li, P., Huang, G., Feng, S., Shen, C., Han, B., Zhang, X., Jin, F., Xu, F., Lu, T.J.: Bioinspired engineering of honeycomb structure —using nature to inspire human innovation. Prog. Mater. Sci. 74, 332–400 (2015). https://doi.org/10.1016/j.pmatsci.2015.05.001
Zhao, Y., Ge, M., Ma, W.: The effective in-plane elastic properties of hexagonal honeycombs with consideration for geometric nonlinearity. Compos. Struct. 234, 111749 (2020). https://doi.org/10.1016/j.compstruct.2019.111749
Zhao, Z., Zhang, D., Chen, W.: An analytical model of blast resistance for all—metallic sandwich panels subjected to shallow - buried explosives. Int. J. Mech. Mater. Des. (2022). https://doi.org/10.1007/s10999-022-09605-w
Acknowledgements
The work is supported by the Science and Engineering Research Board, India, under the scheme ‘Early Career Research Award’, No: ECR/2018/001638 and, the DST-SERB and VSSC, ISRO of the project titled 'Functionality Enhancement through Design and Development of Advanced Finite Element Algorithms for STR tools' under IMPRINT.IIC (IMP/2019/000276) scheme.
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Kumar, A., Muthu, N. & Narayanan, R.G. Equivalent orthotropic properties of periodic honeycomb structure: strain-energy approach and homogenization. Int J Mech Mater Des 19, 137–163 (2023). https://doi.org/10.1007/s10999-022-09620-x
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DOI: https://doi.org/10.1007/s10999-022-09620-x