Abstract
A unified approach to determine the effective elastic properties of irregular and regular honeycomb (Hc) lattice structures is presented. A micromechanics model of a lattice, based on the fundamental periodic element, is developed using Castigliano’s second theorem to obtain a homogenized strain energy density function that yields all elastic properties of the lattice. The lattice geometry, comprising uniform slender beam elements, is specified by six lattice parameters for irregular lattices and four lattice parameters for regular lattices. These parameters generate Hc or re-entrant honeycomb (RHc) structures with different symmetry properties, which can cater to different design requirements. A comprehensive validation, including comparison with available analytical and full finite element (FE) results for different geometries and dimensions, confirms the accuracy of the proposed approach. The influence of the geometric parameters on the effective lattice properties is clearly revealed, which leads to some novel design insights. The existence of the continuous and accidental L symmetry axis in tuned Hc and RHc lattices (including irregular ones) is brought out for the first time in this analysis. Interestingly, it is observed that high auxeticity coincides with low shear modulus, contrary to the existing postulate of high auxeticity that implies high shear stiffness. It is also shown that auxeticity is a directional property in RHc lattice structures.
References
1.
Lakes
, R.
, 1987
, “Foam Structures With a Negative Poisson’s Ratio
,” Science
, 235
(4792
), pp. 1038
–1040
. 2.
Kolken
, H. M. A.
, and Zadpoor
, A. A.
, 2017
, “Auxetic Mechanical Metamaterials
,” RSC Adv.
, 7
(9
), pp. 5111
–5129
. 3.
Ren
, X.
, Das
, R.
, Tran
, P.
, Ngo
, T. D.
, and Xie
, Y. M.
, 2018
, “Auxetic Metamaterials and Structures: A Review
,” Smart Mater. Struct.
, 27
(2
), p. 023001
. 4.
Surjadi
, J. U.
, Gao
, L.
, Du
, H.
, Li
, X.
, Xiong
, X.
, Fang
, N. X.
, and Lu
, Y.
, 2019
, “Mechanical Metamaterials and Their Engineering Applications
,” Adv. Eng. Mater.
, 21
(3
), p. 1800864
. 5.
Wu
, W.
, Hu
, W.
, Qian
, G.
, Liao
, H.
, Xu
, X.
, and Berto
, F.
, 2019
, “Mechanical Design and Multifunctional Applications of Chiral Mechanical Metamaterials: A Review
,” Mater. Des.
, 180
, p. 107950
. 6.
Madhu Balan
, P.
, Mertens
, A. J.
, and Raju Bahubalendruni
, M. V. A.
, 2023
, “Auxetic Mechanical Metamaterials and Their Futuristic Developments: A State-of-Art Review
,” Mater. Today Commun.
, 34
, p. 105285
. 7.
Prawoto
, Y.
, 2012
, “Seeing Auxetic Materials From the Mechanics Point of View: A Structural Review on the Negative Poisson’s Ratio
,” Comput. Mater. Sci.
, 58
, pp. 140
–153
. 8.
Usta
, F.
, Türkmen
, H. S.
, and Scarpa
, F.
, 2021
, “Low-Velocity Impact Resistance of Composite Sandwich Panels With Various Types of Auxetic and Non-auxetic Core Structures
,” Thin-Walled Struct.
, 163
, p. 107738
. 9.
Najafi
, M.
, Ahmadi
, H.
, and Liaghat
, G.
, 2021
, “Experimental Investigation on Energy Absorption of Auxetic Structures
,” Mater. Today: Proc.
, 34
, pp. 350
–355
. 10.
Chen
, Y.
, Li
, T.
, Scarpa
, F.
, and Wang
, L.
, 2017
, “Lattice Metamaterials With Mechanically Tunable Poisson’s Ratio for Vibration Control
,” Phys. Rev. Appl.
, 7
(2
), p. 024012
. 11.
Wang
, C. Y.
, Wang
, W. W.
, Zhao
, W. Z.
, Wang
, Y.
, and Zhou
, G.
, 2018
, “Structure Design and Multi-objective Optimization of a Novel NPR Bumper System
,” Compos. Part B: Eng.
, 153
, pp. 78
–96
. 12.
Wang
, Y.
, Wang
, L.
, dong Ma
, Z.
, and Wang
, T.
, 2016
, “A Negative Poisson’s Ratio Suspension Jounce Bumper
,” Mater. Des.
, 103
, pp. 90
–99
. 13.
Zheng
, Z.
, Rakheja
, S.
, and Sedaghati
, R.
, 2023
, “Modal Properties of Honeycomb Wheels: A Parametric Analysis Using Response Surface Method
,” Eur. J. Mech. A Solids
, 97
, p. 104842
. 14.
Lakes
, R. S.
, and Elms
, K.
, 1993
, “Indentability of Conventional and Negative Poisson’s Ratio Foams
,” J. Compos. Mater.
, 27
(12
), pp. 1193
–1202
. 15.
Alderson
, A.
, Rasburn
, J.
, Ameer-Beg
, S.
, Mullarkey
, P. G.
, Perrie
, W.
, and Evans
, K. E.
, 2000
, “An Auxetic Filter: A Tuneable Filter Displaying Enhanced Size Selectivity Or Defouling Properties
,” Ind. Eng. Chem. Res.
, 39
(3
), pp. 654
–665
. 16.
Evans
, K. E.
, and Alderson
, A.
, 2000
, “Auxetic Materials: Functional Materials and Structures From Lateral Thinking!
” Adv. Mater.
, 12
(9
), pp. 617
–628
. 17.
Carneiro
, V. H.
, Meireles
, J.
, and Puga
, H.
, 2013
, “Auxetic Materials—A Review
,” Mater. Sci. Pol.
, 31
(4
), pp. 561
–571
. 18.
Li
, T.
, Liu
, F.
, and Wang
, L.
, 2020
, “Enhancing Indentation and Impact Resistance in Auxetic Composite Materials
,” Compos. Part B: Eng.
, 198
, p. 108229
. 19.
Kolken
, H. M. A.
, Janbaz
, S.
, Leeflang
, S. M. A.
, Lietaert
, K.
, Weinans
, H. H.
, and Zadpoor
, A. A.
, 2018
, “Rationally Designed Meta-implants: A Combination of Auxetic and Conventional Meta-biomaterials
,” Mater. Horiz.
, 5
(1
), pp. 28
–35
. 20.
Shirzad
, M.
, Zolfagharian
, A.
, Bodaghi
, M.
, and Nam
, S. Y.
, 2023
, “Auxetic Metamaterials for Bone-Implanted Medical Devices: Recent Advances and New Perspectives
,” Eur. J. Mech. A Solids
, 98
, p. 104905
. 21.
Chen
, H.
, Zhu
, F.
, Jang
, K.-I.
, Feng
, X.
, Rogers
, J. A.
, Zhang
, Y.
, Huang
, Y.
, and Ma
, Y.
, 2018
, “The Equivalent Medium of Cellular Substrate Under Large Stretching, With Applications to Stretchable Electronics
,” J. Mech. Phys. Solids
, 120
, pp. 199
–207
(Special Issue in Honor of Ares J. Rosakis on the Occasion of His 60th Birthday). 22.
Yan
, Z. G.
, Wang
, B. L.
, Wang
, K. F.
, and Zhang
, C.
, 2019
, “A Novel Cellular Substrate for Flexible Electronics With Negative Poisson Ratios Under Large Stretching
,” Int. J. Mech. Sci.
, 151
, pp. 314
–321
. 23.
Zhang
, Z.
, Liu
, S.
, Wu
, M.
, and Liu
, S.
, 2023
, “Shape-Adaptable and Wearable Strain Sensor Based on Braided Auxetic Yarns for Monitoring Large Human Motions
,” Appl. Mater. Today
, 35
, p. 101996
. 24.
Jiang
, Y.
, Liu
, Z.
, Matsuhisa
, N.
, Qi
, D.
, Leow
, W. R.
, Yang
, H.
, Yu
, J.
, et al., 2018
, “Auxetic Mechanical Metamaterials to Enhance Sensitivity of Stretchable Strain Sensors
,” Adv. Mater.
, 30
(12
), p. 1706589
. 25.
Ahmed
, M. F.
, Li
, Y.
, and Zeng
, C.
, 2019
, “Stretchable and Compressible Piezoresistive Sensors From Auxetic Foam and Silver Nanowire
,” Mater. Chem. Phys.
, 229
, pp. 167
–173
. 26.
Malfa
, F. L.
, Puce
, S.
, Rizzi
, F.
, and De Vittorio
, M.
, 2020
, “A Flexible Carbon Nanotubes-Based Auxetic Sponge Electrode for Strain Sensors
,” Nanomaterials
, 10
(12
). 27.
Oh
, J.-H.
, Kim
, J.-S.
, Nguyen
, V. H.
, and Oh
, I.-K.
, 2020
, “Auxetic Graphene Oxide-Porous Foam for Acoustic Wave and Shock Energy Dissipation
,” Compos. Part B: Eng.
, 186
, p. 107817
. 28.
Hosseinkhani
, A.
, Younesian
, D.
, Ranjbar
, M.
, and Scarpa
, F.
, 2021
, “Enhancement of the Vibro-acoustic Performance of Anti-tetra-chiral Auxetic Sandwich Panels Using Topologically Optimized Local Resonators
,” Appl. Acoust.
, 177
, p. 107930
. 29.
Eghbali
, P.
, Younesian
, D.
, and Farhangdoust
, S.
, 2020
, “Enhancement of the Low-Frequency Acoustic Energy Harvesting With Auxetic Resonators
,” Appl. Energy
, 270
, p. 115217
. 30.
Latourte
, F.
, Wei
, X.
, Feinberg
, Z. D.
, de Vaucorbeil
, A.
, Tran
, P.
, Olson
, G. B.
, and Espinosa
, H. D.
, 2012
, “Design and Identification of High Performance Steel Alloys for Structures Subjected to Underwater Impulsive Loading
,” Int. J. Solids Struct.
, 49
(13
), pp. 1573
–1587
. 31.
Wei
, X.
, Tran
, P.
, de Vaucorbeil
, A.
, Ramaswamy
, R. B.
, Latourte
, F.
, and Espinosa
, H. D.
, 2013
, “Three-Dimensional Numerical Modeling of Composite Panels Subjected to Underwater Blast
,” J. Mech. Phys. Solids
, 61
(6
), pp. 1319
–1336
. 32.
Wei
, X.
, de Vaucorbeil
, A.
, Tran
, P.
, and Espinosa
, H. D.
, 2013
, “A New Rate-Dependent Unidirectional Composite Model—Application to Panels Subjected to Underwater Blast
,” J. Mech. Phys. Solids
, 61
(6
), pp. 1305
–1318
. 33.
Lisiecki
, J.
, Błażejewicz
, T.
, Kłysz
, S.
, Gmurczyk
, G.
, Reymer
, P.
, and Mikułowski
, G.
, 2013
, “Tests of Polyurethane Foams With Negative Poisson’s Ratio
,” Phys. Status Solidi (b)
, 250
(10
), pp. 1988
–1995
. 34.
Wang
, Y.-C.
, and Lakes
, R.
, 2002
, “Analytical Parametric Analysis of the Contact Problem of Human Buttocks and Negative Poisson’s Ratio Foam Cushions
,” Int. J. Solids Struct.
, 39
(18
), pp. 4825
–4838
. 35.
Scarpa
, F.
, Giacomin
, J.
, Zhang
, Y.
, and Pastorino
, P.
, 2005
, “Mechanical Performance of Auxetic Polyurethane Foam for Antivibration Glove Applications
,” Cell. Polym.
, 24
(5
), pp. 253
–268
. 36.
Sanami
, M.
, Ravirala
, N.
, Alderson
, K.
, and Alderson
, A.
, 2014
, “Auxetic Materials for Sports Applications
,” Procedia Eng.
, 72
, pp. 453
–458
(The Engineering of Sport 10). 37.
Foster
, L.
, Peketi
, P.
, Allen
, T.
, Senior
, T.
, Duncan
, O.
, and Alderson
, A.
, 2018
, “Application of Auxetic Foam in Sports Helmets
,” Appl. Sci.
, 8
(3
). 38.
Allen
, T.
, Duncan
, O.
, Foster
, L.
, Senior
, T.
, Zampieri
, D.
, Edeh
, V.
, and Alderson
, A.
, 2017
, “Auxetic Foam for Snow-Sport Safety Devices,” Snow Sports Trauma and Safety
, I. S.
Scher
, R. M.
Greenwald
, and N.
Petrone
, eds., Springer International Publishing
, Cham
, pp. 145
–159
.39.
Gibson
, L. J.
, Ashby
, M. F.
, Schajer
, G. S.
, and Robertson
, C. I.
, 1982
, “The Mechanics of Two-Dimensional Cellular Materials
,” Proc. R. Soc. Lond. A Math. Phys. Sci.
, 382
(1782
), pp. 25
–42
. 40.
Kolpakov
, A. G.
, 1985
, “Determination of the Average Characteristics of Elastic Frameworks
,” J. Appl. Math. Mech.
, 49
(6
), pp. 739
–745
. 41.
Bakhvalov
, N.
, and Panasenko
, G.
, 1984
, Averaging of Processes in Periodic Media. Mathematical Problems of the Mechanics of Composites
, Nauka
, Moscow
.42.
Hashin
, Z.
, and Rosen
, B. W.
, 1964
, “The Elastic Moduli of Fiber-Reinforced Materials
,” ASME J. Appl. Mech.
, 31
(2
), pp. 223
–232
. 43.
Theocaris
, P. S.
, Spathis
, G.
, and Sideridis
, E.
, 1982
, “Elastic and Viscoelastic Properties of Fibre-Reinforced Composite Materials
,” Fibre Sci. Technol.
, 17
(3
), pp. 169
–181
. 44.
Gonella
, S.
, and Ruzzene
, M.
, 2008
, “Homogenization and Equivalent In-Plane Properties of Two-Dimensional Periodic Lattices
,” Int. J. Solids Struct.
, 45
(10
), pp. 2897
–2915
. 45.
Masters
, I. G.
, and Evans
, K. E.
, 1996
, “Models for the Elastic Deformation of Honeycombs
,” Compos. Struct.
, 35
(4
), pp. 403
–422
. 46.
Evans
, K. E.
, Nkansah
, M. A.
, and Hutchinson
, I. J.
, 1994
, “Auxetic Foams: Modelling Negative Poisson’s Ratios
,” Acta Metall. Mater.
, 42
(4
), pp. 1289
–1294
. 47.
Lee
, J.
, Choi
, J. B.
, and Choi
, K.
, 1996
, “Application of Homogenization FEM Analysis to Regular and Re-entrant Honeycomb Structures
,” J. Mater. Sci.
, 31
(15
), pp. 4105
–4110
. 48.
Wan
, H.
, Ohtaki
, H.
, Kotosaka
, S.
, and Hu
, G.
, 2004
, “A Study of Negative Poisson’s Ratios in Auxetic Honeycombs Based on a Large Deflection Model
,” Eur. J. Mech. A Solids
, 23
(1
), pp. 95
–106
. 49.
Grima
, J. N.
, Attard
, D.
, Ellul
, B.
, and Gatt
, R.
, 2011
, “An Improved Analytical Model for the Elastic Constants of Auxetic and Conventional Hexagonal Honeycombs
,” Cell. Polym.
, 30
(6
), pp. 287
–310
. 50.
Chen
, D. H.
, and Yang
, L.
, 2011
, “Analysis of Equivalent Elastic Modulus of Asymmetrical Honeycomb
,” Compos. Struct.
, 93
(2
), pp. 767
–773
. 51.
Awasthi
, M.
, Naskar
, S.
, Singh
, A.
, and Mukhopadhyay
, T.
, 2024
, “Constitutive Behavior of Asymmetric Multi-material Honeycombs With Bi-level Variably-Thickened Composite Architecture
,” Thin-Walled Struct.
, 203
, p. 112183
. 52.
Larsen
, U. D.
, Signund
, O.
, and Bouwsta
, S.
, 1997
, “Design and Fabrication of Compliant Micromechanisms and Structures With Negative Poisson’s Ratio
,” J. Microelectromech. Syst.
, 6
(2
), pp. 99
–106
. 53.
Theocaris
, P. S.
, Stavroulakis
, G. E.
, and Panagiotopoulos
, P. D.
, 1997
, “Negative Poisson’s Ratios in Composites With Star-Shaped Inclusions: A Numerical Homogenization Approach
,” Arch. Appl. Mech.
, 67
(4
), pp. 274
–286
. 54.
Wei
, L.
, Zhao
, X.
, Yu
, Q.
, and Zhu
, G.
, 2020
, “A Novel Star Auxetic Honeycomb With Enhanced In-Plane Crushing Strength
,” Thin-Walled Struct.
, 149
, p. 106623
. 55.
Huang
, J.
, Zhang
, Q.
, Scarpa
, F.
, Liu
, Y.
, and Leng
, J.
, 2017
, “In-Plane Elasticity of a Novel Auxetic Honeycomb Design
,” Compos. Part B: Eng.
, 110
, pp. 72
–82
. 56.
Berinskii
, I. E.
, 2018
, “In-Plane Elastic Properties of Auxetic Multilattices
,” Smart Mater. Struct.
, 27
(7
), p. 075012
. 57.
Montgomery-Liljeroth
, E.
, Schievano
, S.
, and Burriesci
, G.
, 2023
, “Elastic Properties of 2d Auxetic Honeycomb Structures—A Review
,” Appl. Mater. Today
, 30
, p. 101722
. 58.
Daniel
, I. M.
, and Ishai
, O.
, 2006
, Engineering Mechanics of Composite Materials
, Oxford University Press
, Oxford, UK
.59.
Gibson
, L. J.
, and Ashby
, M. F.
, 1997
, Cellular Solids: Structure and Properties
, Cambridge University Press
, Cambridge, UK
.Copyright © 2025 by ASME
You do not currently have access to this content.
