Abstract

Negative Poisson’s ratio materials, or auxetics, have drawn attention for the past 30 years. The auxetic effect could lead to improved mechanical properties such as acoustic damping, indentation resistance, or crashworthiness. In this work, two 3D auxetic lattices are introduced. Auxeticity is achieved by design through pre-buckling of the lattice struts. The influence of geometrical parameters on the effective elastic properties is investigated using computational homogenization method with periodic boundary conditions. Effective Young’s modulus is 3D mapped to reveal anisotropy and identify spatial orientations of interest. The effective Poisson ratio is computed for various geometric configurations to characterize auxeticity. Finally, the influence of effective elastic properties on energy dissipation under compression is explored for elastoplastic lattices with different loading directions, using finite element simulations. Results suggest that loading 3D auxetic lattices along their stiffest direction maximizes their crashworthiness.

References

References
1.
Ashby
,
M. F.
, and
Bréchet
,
Y.
,
2003
, “
Designing Hybrid Materials
,”
Acta Mater.
,
51
(
19
), pp.
5801
5821
. 10.1016/S1359-6454(03)00441-5
2.
Bouaziz
,
O.
,
Bréchet
,
Y.
, and
Embury
,
J. D.
,
2008
, “
Heterogeneous and Architectured Materials: A Possible Strategy for Design of Structural Materials
,”
Adv. Eng. Mater.
,
10
(
1–2
), pp.
24
36
. 10.1002/(ISSN)1527-2648
3.
Bréchet
,
Y.
, and
Embury
,
J. D.
,
2013
, “
Architectured Materials: Expanding Materials Space
,”
Scr. Mater.
,
68
(
1
), pp.
1
3
. 10.1016/j.scriptamat.2012.07.038
4.
Dirrenberger
,
J.
,
2018
, “
Towards an Integrated Approach for the Development of Architectured Materials
,” Habilitation thesis, Sorbonne Universitéhttps://hal.sorbonne-universite.fr/tel-02047005.
5.
Dirrenberger
,
J.
,
Forest
,
S.
, and
Jeulin
,
D.
,
2019
, “Computational Homogenization of Architectured Materials,”
Architectured Materials in Nature and Engineering
,
Y.
Estrin
,
Y.
Bréchet
,
J.
Dunlop
, and
P.
Fratzl
, eds., Vol. 282 of Springer Series in Materials Science,
Springer
,
New York
, Chap. 4, pp.
89
139
.
6.
Evans
,
K. E.
,
Nkansah
,
M. A.
,
Hutchinson
,
I. J.
, and
Rogers
,
S. C.
,
1991
, “
Molecular Network Design
,”
Nature
,
353
, p.
124
. 10.1038/353124a0
7.
Herakovich
,
C. T.
,
1984
, “
Composite Laminates With Negative Through-the-Thickness Poisson’s Ratios
,”
J. Compos. Mater.
,
18
(
5
), pp.
447
455
. 10.1177/002199838401800504
8.
Almgren
,
R. F.
,
1985
, “
An Isotropic Three-Dimensional Structure With Poisson’s Ratio=−1
,”
J. Elast.
,
15
(
4
), pp.
427
430
. 10.1007/BF00042531
9.
Lakes
,
R. S.
,
1987
, “
Foam Structures With a Negative Poisson’s Ratio
,”
Science
,
235
(
4792
), pp.
1038
1040
. 10.1126/science.235.4792.1038
10.
Bathurst
,
R. J.
, and
Rothenburg
,
L.
,
1988
, “
Note on a Random Isotropic Granular Material With Negative Poisson’s Ratio
,”
Int. J. Eng. Sci.
,
26
(
4
), pp.
373
383
. 10.1016/0020-7225(88)90116-4
11.
Caddock
,
B. D.
, and
Evans
,
K. E.
,
1989
, “
Microporous Materials With Negative Poisson’s Ratios: I. Microstructure and Mechanical Properties
,”
J. Phys. D Appl. Phys.
,
22
(
12
), pp.
1877
1882
. 10.1088/0022-3727/22/12/012
12.
Lakes
,
R. S.
,
1991
, “
Deformation Mechanisms in Negative Poisson’s Ratio Materials: Structural Aspects
,”
J. Mater. Sci.
,
26
(
9
), pp.
2287
2292
. 10.1007/BF01130170
13.
Milton
,
G. W.
,
1992
, “
Composite Materials With Poisson’s Ratios Close to -1
,”
J. Mech. Phys. Solids
,
40
(
5
), pp.
1105
1137
. 10.1016/0022-5096(92)90063-8
14.
Prall
,
D.
, and
Lakes
,
R. S.
,
1997
, “
Properties of a Chiral Honeycomb With a Poisson’s Ratio of -1
,”
Int. J. Mech. Sci.
,
39
(
3
), pp.
305
314
. 10.1016/S0020-7403(96)00025-2
15.
Yang
,
W.
,
Li
,
Z.-M.
,
Shi
,
W.
,
Xie
,
B.-H.
, and
Yang
,
M.-B.
,
2004
, “
Review on Auxetic Materials
,”
J. Mater. Sci.
,
39
(
10
), pp.
3269
3279
. 10.1023/B:JMSC.0000026928.93231.e0
16.
Hughes
,
T. P.
,
Marmier
,
A.
, and
Evans
,
K. E.
,
2010
, “
Auxetic Frameworks Inspired by Cubic Crystals
,”
Int. J. Solids Struct.
,
47
(
11–12)
, pp.
1469
1476
. 10.1016/j.ijsolstr.2010.02.002
17.
Alderson
,
A.
,
Alderson
,
K. L.
,
Attard
,
D.
,
Evans
,
K. E.
,
Gatt
,
R.
,
Grima
,
J. N.
,
Miller
,
W.
,
Ravirala
,
N.
,
Smith
,
C. W.
, and
Zied
,
K.
,
2010
, “
Elastic Constants of 3-, 4- and 6-Connected Chiral and Anti-Chiral Honeycombs Subject to Uniaxial In-Plane Loading
,”
Compos. Sci. Technol.
,
70
(
7
), pp.
1042
1048
. 10.1016/j.compscitech.2009.07.009
18.
Dirrenberger
,
J.
,
Forest
,
S.
,
Jeulin
,
D.
, and
Colin
,
C.
,
2011
, “
Homogenization of Periodic Auxetic Materials
,”
Procedia Eng.
,
10
, pp.
1847
1852
. 10.1016/j.proeng.2011.04.307
19.
Pasternak
,
E.
, and
Dyskin
,
A.
,
2012
, “
Materials and Structures With Macroscopic Negative Poisson’s Ratio
,”
Int. J. Eng. Sci.
,
52
, pp.
103
114
. 10.1016/j.ijengsci.2011.11.006
20.
Alvarez Elipe
,
J. C.
, and
Diaz Lantada
,
A.
,
2012
, “
Comparative Study of Auxetic Geometries by Means of Computer-Aided Design and Engineering
,”
Smart Mater. Struct.
,
21
, p.
105004
. 10.1088/0964-1726/21/10/105004
21.
Dirrenberger
,
J.
,
Forest
,
S.
, and
Jeulin
,
D.
,
2012
, “
Elastoplasticity of Auxetic Materials
,”
Comput. Mater. Sci.
,
64
, pp.
57
61
. 10.1016/j.commatsci.2012.03.036
22.
Dirrenberger
,
J.
,
Forest
,
S.
, and
Jeulin
,
D.
,
2013
, “
Effective Elastic Properties of Auxetic Microstructures: Anisotropy and Structural Applications
,”
Int. J. Mech. Mater. Design
,
9
(
1
), pp.
21
33
. 10.1007/s10999-012-9192-8
23.
Krasavin
,
V.
, and
Krasavin
,
A.
,
2014
, “
Auxetic Properties of Cubic Metal Single Crystals
,”
Phys. Status Solidi (b)
,
251
(
11
), pp.
2314
2320
. 10.1002/pssb.201451129
24.
Kaminakis
,
N.
,
Drosopoulos
,
G.
, and
Stavroulakis
,
G.
,
2015
, “
Design and Verification of Auxetic Microstructures Using Topology Optimization and Homogenization
,”
Arch. Appl. Mech.
,
85
(
9
), pp.
1289
1306
. 10.1007/s00419-014-0970-7
25.
Körner
,
C.
, and
Liebold-Ribeiro
,
Y.
,
2015
, “
A Systematic Approach to Identify Cellular Auxetic Materials
,”
Smart Mater. Struct.
,
24
(
2
), p.
025013
. 10.1088/0964-1726/24/2/025013
26.
Saxena
,
K. K.
,
Das
,
R.
, and
Calius
,
E. P.
,
2016
, “
Three Decades of Auxetics Research— Materials With Negative Poisson’s Ratio: A Review
,”
Adv. Eng. Mater.
,
18
(
11
), pp.
1847
1870
. 10.1002/adem.v18.11
27.
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
. 10.1088/1361-665X/aaa61c
28.
Evans
,
K. E.
,
1991
, “
The Design of Doubly Curved Sandwich Panels With Honeycomb Cores
,”
Compos. Struct.
,
17
(
2
), pp.
95
111
. 10.1016/0263-8223(91)90064-6
29.
Choi
,
J. B.
, and
Lakes
,
R. S.
,
1991
, “
Design of a Fastener Based on Negative Poisson’s Ratio Foam
,”
Cell. Polym.
,
10
(
3
), pp.
205
212
.
30.
Martin
,
J.
,
Heyder-Bruckner
,
J.-J.
,
Remillat
,
C.
,
Scarpa
,
F.
,
Potter
,
K.
, and
Ruzzene
,
M.
,
2008
, “
The Hexachiral Prismatic Wingbox Concept
,”
Physica Status Solidi (b)
,
245
(
3
), pp.
570
577
. 10.1002/(ISSN)1521-3951
31.
Bertoldi
,
K.
,
Reis
,
P.
,
Willshaw
,
S.
, and
Mullin
,
T.
,
2010
, “
Negative Poisson’s Ratio Behavior Induced by An Elastic Instability
,”
Adv. Mater.
,
22
(
3
), pp.
361
366
. 10.1002/adma.v22:3
32.
Agnese
,
F.
,
Remillat
,
C.
,
Scarpa
,
F.
, and
Payne
,
C.
,
2015
, “
Composite Chiral Shear Vibration Damper
,”
Compos. Struct.
,
132
, pp.
215
225
. 10.1016/j.compstruct.2015.05.048
33.
Jang
,
K.-I.
,
Chung
,
H. U.
,
Xu
,
S.
,
Lee
,
C. H.
,
Luan
,
H.
,
Jeong
,
J.
,
Cheng
,
H.
,
Kim
,
G.-T.
,
Han
,
S. Y.
,
Lee
,
J. W.
,
Kim
,
J.
,
Cho
,
M.
,
Miao
,
F.
,
Yang
,
Y.
,
Jung
,
H. N.
,
Flavin
,
M.
,
Liu
,
H.
,
Kong
,
G. W.
,
Yu
,
K. J.
,
Rhee
,
S. I.
,
Chung
,
J.
,
Kim
,
B.
,
Kwak
,
J. W.
,
Yun
,
M. H.
,
Kim
,
J. Y.
,
Song
,
Y. M.
,
Paik
,
U.
,
Zhang
,
Y.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2015
, “
Soft Network Composite Materials With Deterministic and Bio-inspired Designs
,”
Nat. Commun.
,
6
, p.
6566
. 10.1038/ncomms7566
34.
Lipsett
,
A. W.
, and
Beltzer
,
A. I.
,
1988
, “
Reexamination of Dynamic Problems of Elasticity for Negative Poisson’s Ratio
,”
J. Acoust. Soc. Am.
,
84
(
6
), pp.
2179
2186
. 10.1121/1.397064
35.
Howell
,
B.
,
Prendergast
,
P.
, and
Hansen
,
L.
,
1994
, “
Examination of Acoustic Behavior of Negative Poisson’s Ratio Materials
,”
Appl. Acoust.
,
43
(
2
), pp.
141
148
. 10.1016/0003-682x(94)90057-4
36.
Chen
,
C. P.
, and
Lakes
,
R. S.
,
1996
, “
Micromechanical Analysis of Dynamic Behavior of Conventional and Negative Poisson’s Ratio Foams
,”
ASME J. Eng. Mater. Technol.
,
118
(
3
), pp.
285
288
. 10.1115/1.2806807
37.
Scarpa
,
F.
,
Ciffo
,
L. G.
, and
Yates
,
J. R.
,
2004
, “
Dynamic Properties of High Structural Integrity Auxetic Open Cell Foam
,”
Smart Mater. Struct.
,
13
(
1
), p.
49
.
38.
Chekkal
,
I.
,
Bianchi
,
M.
,
Remillat
,
C.
,
Becot
,
F.-X.
,
Jaouen
,
L.
, and
Scarpa
,
F.
,
2010
, “
Vibro-Acoustic Properties of Auxetic Open Cell Foam: Model and Experimental Results
,”
Acta Acustica United Acustica
,
96
(
2
), pp.
266
274
. 10.3813/AAA.918276
39.
Spadoni
,
A.
,
Ruzzene
,
M.
,
Gonella
,
S.
, and
Scarpa
,
F.
,
2009
, “
Phononic Properties of Hexagonal Chiral Lattices
,”
Wave Motion
,
46
(
7
), pp.
435
450
. 10.1016/j.wavemoti.2009.04.002
40.
Auffray
,
N.
,
Dirrenberger
,
J.
, and
Rosi
,
G.
,
2015
, “
A Complete Description of Bi-dimensional Anisotropic Strain-Gradient Elasticity
,”
Int. J. Solids. Struct.
,
69–70
, pp.
195
210
. 10.1016/j.ijsolstr.2015.04.036
41.
Rosi
,
G.
, and
Auffray
,
N.
,
2016
, “
Anisotropic and Dispersive Wave Propagation Within Strain-Gradient Framework
,”
Wave Motion
,
63
, pp.
120
134
. 10.1016/j.wavemoti.2016.01.009
42.
Poncelet
,
M.
,
Somera
,
A.
,
Morel
,
C.
,
Jailin
,
C.
, and
Auffray
,
N.
,
2018
, “
An Experimental Evidence of the Failure of Cauchy Elasticity for the Overall Modeling of a Non-Centro-Symmetric Lattice Under Static Loading
,”
Int. J. Solids Struct.
,
147
, pp.
223
237
. 10.1016/j.ijsolstr.2018.05.028
43.
Scarpa
,
F.
,
Yates
,
J. R.
,
Ciffo
,
L. G.
, and
Patsias
,
S.
,
2002
, “
Dynamic Crushing of Auxetic Open-Cell Polyurethane Foam
,”
Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.
,
216
(
12
), pp.
1153
1156
. 10.1243/095440602321029382
44.
Liu
,
W.
,
Wang
,
N.
,
Luo
,
T.
, and
Lin
,
Z.
,
2016
, “
In-Plane Dynamic Crushing of Re-Entrant Auxetic Cellular Structure
,”
Mater. Des.
,
100
, pp.
84
91
. 10.1016/j.matdes.2016.03.086
45.
Li
,
T.
,
Chen
,
Y.
,
Hu
,
X.
,
Li
,
Y.
, and
Wang
,
L.
,
2018
, “
Exploiting Negative Poisson’s Ratio to Design 3d-Printed Composites With Enhanced Mechanical Properties
,”
Mater. Des.
,
142
, pp.
247
258
. 10.1016/j.matdes.2018.01.034
46.
Novak
,
N.
,
Starčevič
,
L.
,
Vesenjak
,
M.
, and
Ren
,
Z.
,
2019
, “
Blast Response Study of the Sandwich Composite Panels With 3d Chiral Auxetic Core
,”
Composite Structures
,
210
, pp.
167
178
. 10.1016/j.compstruct.2018.11.050
47.
Chan
,
N.
, and
Evans
,
K. E.
,
1998
, “
Indentation Resilience of Conventional and Auxetic Foams
,”
J. Cell. Plast.
,
34
(
3
), pp.
231
260
. 10.1177/0021955X9803400304
48.
Gilat
,
R.
, and
Aboudi
,
J.
,
2013
, “
Behavior of Elastoplastic Auxetic Microstructural Arrays
,”
Materials
,
6
(
3
), pp.
726
737
. 10.3390/ma6030726
49.
Ghaedizadeh
,
A.
,
Shen
,
J.
,
Ren
,
X.
, and
Xie
,
Y. M.
,
2016
, “
Tuning the Performance of Metallic Auxetic Metamaterials by Using Buckling and Plasticity
,”
Materials
,
9
(
54
), pp.
1
17
.
50.
Bornert
,
M.
,
Bretheau
,
T.
, and
Gilormini
,
P.
,
2001
,
Homogénéisation En Mécanique Des Matériaux, Tome 1 : Matériaux Aléatoires élastiques Et Milieux Périodiques
,
Hermès
,
Paris
.
51.
Besson
,
J.
,
Cailletaud
,
G.
,
Chaboche
,
J.-L.
,
Forest
,
S.
, and
Blétry
,
M.
,
2010
,
Non-Linear Mechanics of Materials
, Vol. 167 of Solid Mechanics and Its Applications,
Springer
,
New York
.
52.
Jiang
,
M.
,
Jasiuk
,
I.
, and
Ostoja-Starzewski
,
M.
,
2002
, “
Apparent Elastic and Elastoplastic Behavior of Periodic Composites
,”
Int. J. Solids Struct.
,
39
(
1
), pp.
199
212
. 10.1016/S0020-7683(01)00145-7
53.
Cailletaud
,
G.
,
Forest
,
S.
,
Jeulin
,
D.
,
Feyel
,
F.
,
Galliet
,
I.
,
Mounoury
,
V.
, and
Quilici
,
S.
,
2003
, “
Some Elements of Microstructural Mechanics
,”
Comput. Mater. Sci.
,
27
(
3
), pp.
351
374
. 10.1016/S0927-0256(03)00041-7
54.
Yuan
,
Z.
, and
Fish
,
J.
,
2008
, “
Toward Realization of Computational Homogenization in Practice
,”
Int. J. Numer. Methods Eng.
,
73
(
3
), pp.
361
380
. 10.1002/(ISSN)1097-0207
55.
Fritzen
,
F.
,
Forest
,
S.
,
Kondo
,
D.
, and
Böhlke
,
T.
,
2013
, “
Computational Homogenization of Porous Materials of Green Type
,”
Computat. Mech.
,
52
(
1
), pp.
121
134
. 10.1007/s00466-012-0801-z
56.
Geers
,
M. G. D.
, and
Yvonnet
,
J.
,
2016
, “
Multiscale Modeling of Microstructure-Property Relations
,”
MRS Bull.
,
41
(
8
), pp.
610
616
. 10.1557/mrs.2016.165
57.
Evans
,
A. G.
,
Hutchinson
,
J. W.
,
Fleck
,
N. A.
,
Ashby
,
M. F.
, and
Wadley
,
H. N. G.
,
2001
, “
The Topological Design of Multifunctional Cellular Metals
,”
Prog. Mater. Sci.
,
46
(
3–4
), pp.
309
327
. 10.1016/S0079-6425(00)00016-5
58.
Deshpande
,
V. S.
,
Ashby
,
M. F.
, and
Fleck
,
N. A.
,
2001
, “
Foam Topology: Bending Versus Stretching Dominated Architectures
,”
Acta Mater.
,
49
(
6
), pp.
1035
1040
. 10.1016/S1359-6454(00)00379-7
59.
Deshpande
,
V. S.
,
Fleck
,
N. A.
, and
Ashby
,
M. F.
,
2001
, “
Effective Properties of the Octet-Truss Lattice Material
,”
J. Mech. Phys. Solids
,
49
(
8
), pp.
1747
1769
. 10.1016/S0022-5096(01)00010-2
60.
Kooistra
,
G. W.
,
Deshpande
,
V. S.
, and
Wadley
,
H. N.
,
2004
, “
Compressive Behavior of Age Hardenable Tetrahedral Lattice Truss Structures Made From Aluminium
,”
Acta Mater.
,
52
(
14
), pp.
4229
4237
. 10.1016/j.actamat.2004.05.039
61.
Ashby
,
M. F.
,
2006
, “
The Properties of Foams and Lattices
,”
Philos. Trans. R. Soc. A Math., Phys. Eng. Sci.
,
364
(
1838
), pp.
15
30
. 10.1098/rsta.2005.1678
62.
Fleck
,
N. A.
,
Deshpande
,
V. S.
, and
Ashby
,
M. F.
,
2010
, “
Micro-architectured Materials: Past, Present and Future
,”
Proc. R. Soc. A Math., Phys. Eng. Sci.
,
466
(
2121
), pp.
2495
2516
. 10.1098/rspa.2010.0215
63.
Vigliotti
,
A.
, and
Pasini
,
D.
,
2012
, “
Stiffness and Strength of Tridimensional Periodic Lattices
,”
Comput. Methods Appl. Mech. Eng.
,
229
, pp.
27
43
. 10.1016/j.cma.2012.03.018
64.
Vigliotti
,
A.
, and
Pasini
,
D.
,
2013
, “
Mechanical Properties of Hierarchical Lattices
,”
Mech. Mater.
,
62
, pp.
32
43
. 10.1016/j.mechmat.2013.03.003
65.
Zok
,
F. W.
,
Latture
,
R. M.
, and
Begley
,
M. R.
,
2016
, “
Periodic Truss Structures
,”
J. Mech. Phys. Solids
,
96
, pp.
184
203
. 10.1016/j.jmps.2016.07.007
66.
Latture
,
R. M.
,
Begley
,
M. R.
, and
Zok
,
F. W.
,
2018
, “
Design and Mechanical Properties of Elastically Isotropic Trusses
,”
J. Mater. Res.
,
33
(
3
), pp.
249
263
. 10.1557/jmr.2018.2
67.
Latture
,
R. M.
,
Rodriguez
,
R. X.
,
Holmes
,
L. R.
, Jr.
, and
Zok
,
F. W.
,
2018
, “
Effects of Nodal Fillets and External Boundaries on Compressive Response of an Octet Truss
,”
Acta Mater.
,
149
, pp.
78
87
. 10.1016/j.actamat.2017.12.060
68.
Tancogne-Dejean
,
T.
, and
Mohr
,
D.
,
2018
, “
Elastically-Isotropic Truss Lattice Materials of Reduced Plastic Anisotropy
,”
Int. J. Solids Struct.
,
138
, pp.
24
39
. 10.1016/j.ijsolstr.2017.12.025
69.
Bonatti
,
C.
, and
Mohr
,
D.
,
2019
, “
Mechanical Performance of Additively-Manufactured Anisotropic and Isotropic Smooth Shell-Lattice Materials: Simulations & Experiments
,”
J. Mech. Phys. Solids
,
122
, pp.
1
26
. 10.1016/j.jmps.2018.08.022
70.
Maxwell
,
J. C.
,
1864
, “
On the Calculation of the Equilibrium and Stiffness of Frames
,”
London, Edinburgh, Dublin Philos. Mag. J. Sci.
,
27
(
182
), pp.
294
299
. 10.1080/14786446408643668
71.
Calladine
,
C. R.
,
1978
, “
Buckminster Fuller’s ‘Tensegrity’ Structures and Clerk Maxwell’s Rules for the Construction of Stiff Frames
,”
Int. J. Solids Struct.
,
14
(
2
), pp.
161
172
. 10.1016/0020-7683(78)90052-5
72.
Pellegrino
,
S.
, and
Calladine
,
C. R.
,
1986
, “
Matrix Analysis of Statically and Kinematically Indeterminate Frameworks
,”
Int. J. Solids Struct.
,
22
(
4
), pp.
409
428
. 10.1016/0020-7683(86)90014-4
73.
Mitschke
,
H.
,
Schury
,
F.
,
Mecke
,
K.
,
Wein
,
F.
,
Stingl
,
M.
, and
Schröder-Turk
,
G.
,
2016
, “
Geometry: The Leading Parameter for the Poissons Ratio of Bending-Dominated Cellular Solids
,”
Int. J. Solids Struct.
,
100–101
, pp.
1
10
. 10.1016/j.ijsolstr.2016.06.027
74.
Rayneau-Kirkhope
,
D.
,
2018
, “
Stiff Auxetics: Hierarchy as a Route to Stiff, Strong Lattice Based Auxetic Meta-materials
,”
Sci. Rep.
,
8
, p.
12437
. 10.1038/s41598-018-29025-1
75.
Weaver
,
P. M.
, and
Ashby
,
M. F.
,
1996
, “
The Optimal Selection of Material and Section-Shape
,”
J. Eng. Design
,
7
(
2
), pp.
129
150
. 10.1080/09544829608907932
76.
Gibson
,
L. J.
, and
Ashby
,
M. F.
,
1999
,
Cellular Materials
,
Cambridge University Press
,
Cambridge
.
77.
Ashby
,
M.
,
2013
, “
Designing Architectured Materials
,”
Scr. Mater.
,
68
(
1
), pp.
4
7
. 10.1016/j.scriptamat.2012.04.033
78.
Warmuth
,
F.
,
Wormser
,
M.
, and
Körner
,
C.
,
2017
, “
Single Phase 3d Phononic Band Gap Material
,”
Sci. Rep.
,
7
(
1
), p.
3843
. 10.1038/s41598-017-04235-1
79.
Bertoldi
,
K.
,
Reis
,
P. M.
,
Willshaw
,
S.
, and
Mullin
,
T.
,
2010
, “
Negative Poisson’s Ratio Behavior Induced by an Elastic Instability
,”
Adv. Mater.
,
22
(
3
), pp.
361
366
. 10.1002/adma.200901956
80.
Babaee
,
S.
,
Shim
,
J.
,
Weaver
,
J.
,
Chen
,
E.
,
Patel
,
N.
, and
Bertoldi
,
K.
,
2013
, “
3D Soft Metamaterials With Negative Poisson’s Ratio
,”
Adv. Mater.
,
25
(
36
), pp.
5044
5049
. 10.1002/adma.201301986
81.
Ren
,
X.
,
Shen
,
J.
,
Ghaedizadeh
,
A.
,
Tian
,
H.
, and
Xie
,
Y. M.
,
2015
, “
Experiments and Parametric Studies on 3d Metallic Auxetic Metamaterials With Tuneable Mechanical Properties
,”
Smart Materials Struct.
,
24
(
9
), p.
095016
. 10.1088/0964-1726/24/9/095016
82.
Ren
,
X.
,
Shen
,
J.
,
Tran
,
P.
,
Ngo
,
T. D.
, and
Xie
,
Y. M.
,
2018
, “
Design and Characterisation of a Tuneable 3d Buckling-Induced Auxetic Metamaterial
,”
Mater. Des.
,
139
, pp.
336
342
. 10.1016/j.matdes.2017.11.025
83.
Bacigalupo
,
A.
, and
Gambarotta
,
L.
,
2014
, “
Homogenization of Periodic Hexa- and Tetrachiral Cellular Solids
,”
Compos. Struct.
,
116
(
1
), pp.
461
476
. 10.1016/j.compstruct.2014.05.033
84.
Bacigalupo
,
A.
, and
De Bellis
,
M. L.
,
2015
, “
Auxetic Anti-tetrachiral Materials: Equivalent Elastic Properties and Frequency Band-Gaps
,”
Compos. Struct.
,
131
, pp.
530
544
. 10.1016/j.compstruct.2015.05.039
85.
Mukhopadhyay
,
T.
, and
Adhikari
,
S.
,
2016
, “
Effective In-Plane Elastic Properties of Auxetic Honeycombs With Spatial Irregularity
,”
Mech. Mater.
,
95
, pp.
204
222
. 10.1016/j.mechmat.2016.01.009
86.
Geuzaine
,
C.
, and
Remacle
,
J.-F.
,
2009
, “
Gmsh: A 3-D Finite Element Mesh Generator With Built-In Pre- and Post-Processing Facilities
,”
Int. J. Numer. Meth. Eng.
,
79
(
11
), pp.
1309
1331
. 10.1002/nme.2579
87.
Sab
,
K.
,
1992
, “
On the Homogenization and the Simulation of Random Materials
,”
Eur. J. Mech., A/Solids
,
11
(
5
), pp.
585
607
.
88.
Kanit
,
T.
,
Forest
,
S.
,
Galliet
,
I.
,
Mounoury
,
V.
, and
Jeulin
,
D.
,
2003
, “
Determination of the Size of the Representative Volume Element for Random Composites: Statistical and Numerical Approach
,”
Int. J. Solids Struct.
,
40
(
13–14
), pp.
3647
3679
. 10.1016/S0020-7683(03)00143-4
89.
Bunge
,
H.-J.
,
1982
,
Texture Analysis in Materials Science: Mathematical Methods
,
Elsevier
,
New York
,
90.
Pouzet
,
S.
,
Peyre
,
P.
,
Gorny
,
C.
,
Castelnau
,
O.
,
Baudin
,
T.
,
Brisset
,
F.
,
Colin
,
C.
, and
Gadaud
,
P.
,
2016
, “
Additive Layer Manufacturing of Titanium Matrix Composites Using the Direct Metal Deposition Laser Process
,”
Mater. Sci. Eng. A.
,
677
, pp.
171
181
. 10.1016/j.msea.2016.09.002
91.
Warmuth
,
F.
,
Osmanlic
,
F.
,
Adler
,
L.
,
Lodes
,
M. A.
, and
Körner
,
C.
,
2016
, “
Fabrication and Characterisation of a Fully Auxetic 3d Lattice Structure Via Selective Electron Beam Melting
,”
Smart Mater. Struct.
,
26
(
2
), p.
025013
. 10.1088/1361-665X/26/2/025013
92.
Auffray
,
N.
,
2014
, “
Analytical Expressions for Odd-Order Anisotropic Tensor Dimension
,”
Comptes Rendus Mécaniques
,
342
(
5
), pp.
284
291
. 10.1016/j.crme.2014.01.012
93.
Auffray
,
N.
,
He
,
Q. C.
, and
Le Quang
,
H.
,
2019
, “
Complete Symmetry Classification and Compact Matrix Representations for 3D Strain Gradient Elasticity
,”
Int. J. Solids Struct.
,
159
, pp.
197
210
. 10.1016/j.ijsolstr.2018.09.029
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