Abstract

Cellular materials are critical elements for mechanical metamaterials design and fabrication. Tailoring the internal cellular material structural pattern can achieve a much broader range of bulk properties than the constituent materials, thus enabling the metamaterial design with extraordinary properties. Studying cellular materials’ mechanical properties is critical for understanding metamaterials’ structural design, and macroscale performances. This paper investigates and validates the mechanical properties of two classes of smooth cellular structures defined by deterministic and stochastic functions, respectively. A mechanical profile is proposed to depict the effective mechanical properties of these smooth cellular structures. We developed such profiles numerically based on computational homogenization and validated them by simulations and physical tests. In physical tests, we printed the generated structures on a fused deposition modeling (FDM) printer and conducted compression tests to verify the numerical homogenization and simulation results. Through the comparison studies, we summarize these cellular materials’ mechanical profiles defined by distinct principles. Based on the experimental results, several cellular structural design guidelines are derived for mechanical metamaterials development, which provides foundations for cellular materials database establishment and sheds light on future exotic metamaterials fabrication.

References

1.
Barchiesi
,
E.
,
Spagnuolo
,
M.
, and
Placidi
,
L.
,
2019
, “
Mechanical Metamaterials: A State of the Art
,”
Math. Mech. Solids
,
24
(
1
), pp.
212
234
.
2.
Lee
,
J.-H.
,
Singer
,
J. P.
, and
Thomas
,
E. L.
,
2012
, “
Micro-/Nanostructured Mechanical Metamaterials
,”
Adv. Mater.
,
24
(
36
), pp.
4782
4810
.
3.
Bertoldi
,
K.
,
Vitelli
,
V.
,
Christensen
,
J.
, and
Van Hecke
,
M.
,
2017
, “
Flexible Mechanical Metamaterials
,”
Nat. Rev. Mater.
,
2
(
11
), pp.
1
11
.
4.
Zadpoor
,
A. A.
,
2016
, “
Mechanical Meta-Materials
,”
Mater. Horiz.
,
3
(
5
), pp.
371
381
.
5.
Zheng
,
X.
,
Lee
,
H.
,
Weisgraber
,
T. H.
,
Shusteff
,
M.
,
DeOtte
,
J.
,
Duoss
,
E. B.
,
Kuntz
,
J. D.
,
Biener
,
M. M.
,
Ge
,
Q.
,
Jackson
,
J. A.
, and
Kucheyev
,
S. O.
,
2014
, “
Ultralight, Ultrastiff Mechanical Metamaterials
,”
Science
,
344
(
6190
), pp.
1373
1377
.
6.
Bauer
,
J.
,
Meza
,
L. R.
,
Schaedler
,
T. A.
,
Schwaiger
,
R.
,
Zheng
,
X.
, and
Valdevit
,
L.
,
2017
, “
Nanolattices: An Emerging Class of Mechanical Metamaterials
,”
Adv. Mater.
,
29
(
40
), p.
1701850
.
7.
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
.
8.
Wang
,
Q.
,
Jackson
,
J. A.
,
Ge
,
Q.
,
Hopkins
,
J. B.
,
Spadaccini
,
C. M.
, and
Fang
,
N. X.
,
2016
, “
Lightweight Mechanical Metamaterials With Tunable Negative Thermal Expansion
,”
Phys. Rev. Lett.
,
117
(
17
), p.
175901
.
9.
Haberman
,
M. R.
, and
Guild
,
M. D.
,
2016
, “
Acoustic Metamaterials
,”
Phys. Today
,
69
(
6
), pp.
42
48
.
10.
Sihvola
,
A.
,
2007
, “
Metamaterials in Electromagnetics
,”
Metamaterials
,
1
(
1
), pp.
2
11
.
11.
McKittrick
,
J.
,
Chen
,
P.-Y.
,
Tombolato
,
L.
,
Novitskaya
,
E.
,
Trim
,
M.
,
Hirata
,
G.
,
Olevsky
,
E.
,
Horstemeyer
,
M.
, and
Meyers
,
M.
,
2010
, “
Energy Absorbent Natural Materials and Bioinspired Design Strategies: A Review
,”
Mater. Sci. Eng.: C
,
30
(
3
), pp.
331
342
.
12.
Zhu
,
J.-H.
,
Zhang
,
W.-H.
, and
Xia
,
L.
,
2016
, “
Topology Optimization in Aircraft and Aerospace Structures Design
,”
Arch. Comput. Methods Eng.
,
23
(
4
), pp.
595
622
.
13.
Park
,
J.
,
Zobaer
,
T.
, and
Sutradhar
,
A.
,
2022
, “
A Two-Scale Multi-Resolution Topologically Optimized Multi-Material Design of 3D Printed Craniofacial Bone Implants
,”
Micromachines
,
12
(
2
), p.
101
.
14.
Wang
,
J.
, and
Huang
,
J.
,
2022
, “
Functionally Graded Non-Periodic Cellular Structure Design and Optimization
,”
ASME J. Comput. Inf. Sci. Eng.
,
22
(
3
), p.
031006
.
15.
Richard
,
C. T.
, and
Kwok
,
T. -H.
,
2021
, “
Analysis and Design of Lattice Structures for Rapid-Investment Casting
,”
Materials
,
14
(
17
), p.
4867
.
16.
Portela
,
C. M.
,
Greer
,
J. R.
, and
Kochmann
,
D. M.
,
2018
, “
Impact of Node Geometry on the Effective Stiffness of Non-Slender Three-Dimensional Truss Lattice Architectures
,”
Extreme Mech. Lett.
,
22
, pp.
138
148
.
17.
Mateos
,
A. J.
,
Huang
,
W.
,
Zhang
,
Y. -W.
, and
Greer
,
J. R.
,
2019
, “
Discrete-Continuum Duality of Architected Materials: Failure, Flaws, and Fracture
,”
Adv. Funct. Mater.
,
29
(
5
), p.
1806772
.
18.
Latture
,
R. M.
,
Rodriguez
,
R. X.
,
Holmes Jr
,
L. R.
, and
Zok
,
F. W.
,
2018
, “
Effects of Nodal Fillets and External Boundaries on Compressive Response of An Octet Truss
,”
Acta. Mater.
,
149
(
47
), pp.
78
87
.
19.
Gandy
,
P. J.
,
Bardhan
,
S.
,
Mackay
,
A. L.
, and
Klinowski
,
J.
,
2001
, “
Nodal Surface Approximations to the P, G, D and I-WP Triply Periodic Minimal Surfaces
,”
Chem. Phys. Lett.
,
336
(
3–4
), pp.
187
195
.
20.
Meeks III
,
W.
, and
Pérez
,
J.
,
2011
, “
The Classical Theory of Minimal Surfaces
,”
Bull. Am. Math. Soc.
,
48
(
3
), pp.
325
407
.
21.
Hsieh
,
M.-T.
,
Endo
,
B.
,
Zhang
,
Y.
,
Bauer
,
J.
, and
Valdevit
,
L.
,
2019
, “
The Mechanical Response of Cellular Materials With Spinodal Topologies
,”
J. Mech. Phys. Solids.
,
125
, pp.
401
419
.
22.
Rajagopalan
,
S.
, and
Robb
,
R. A.
,
2006
, “
Schwarz Meets Schwann: Design and Fabrication of Biomorphic and Durataxic Tissue Engineering Scaffolds
,”
Med. Image Anal.
,
10
(
5
), pp.
693
712
.
23.
Liu
,
P.
,
Liu
,
A.
,
Peng
,
H.
,
Tian
,
L.
,
Liu
,
J.
, and
Lu
,
L.
,
2021
, “
Mechanical Property Profiles of Microstructures Via Asymptotic Homogenization
,”
Comput. Graph.
,
100
(
C
), pp.
106
115
.
24.
Von Schnering
,
H.
, and
Nesper
,
R.
,
1991
, “
Nodal Surfaces of Fourier Series: Fundamental Invariants of Structured Matter
,”
Zeitschrift für Physik B Condensed Matter
,
83
(
3
), pp.
407
412
.
25.
Rastegarzadeh
,
S.
,
Wang
,
J.
, and
Huang
,
J.
,
2021
, “
Two-Scale Topology Optimization With Parameterized Cellular Structures
,”
International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Vol.
85376
,
Virtual online
,
Aug. 19–21
,
American Society of Mechanical Engineers
, p. V002T02A046.
26.
Fritzen
,
F.
,
Böhlke
,
T.
, and
Schnack
,
E.
,
2009
, “
Periodic Three-Dimensional Mesh Generation for Crystalline Aggregates Based on Voronoi Tessellations
,”
Comput. Mech.
,
43
(
5
), pp.
701
713
.
27.
Kumar
,
S.
,
Tan
,
S.
,
Zheng
,
L.
, and
Kochmann
,
D. M.
,
2020
, “
Inverse-Designed Spinodoid Metamaterials
,”
npj Comput. Mater.
,
6
(
1
), pp.
1
10
.
28.
Cahn
,
J. W.
,
1965
, “
Phase Separation by Spinodal Decomposition in Isotropic Systems
,”
J. Chem. Phys.
,
42
(
1
), pp.
93
99
.
29.
Soyarslan
,
C.
,
Bargmann
,
S.
,
Pradas
,
M.
, and
Weissmüller
,
J.
,
2018
, “
3D Stochastic Bicontinuous Microstructures: Generation, Topology and Elasticity
,”
Acta. Mater.
,
149
, pp.
326
340
.
30.
Dong
,
G.
,
Tang
,
Y.
, and
Zhao
,
Y. F.
,
2019
, “
A 149 Line Homogenization Code for Three-Dimensional Cellular Materials Written in MATLAB
,”
ASME J. Eng. Mater. Technol.
,
141
(
1
), p.
011005
.
31.
Hassani
,
B.
, and
Hinton
,
E.
,
2012
,
Homogenization and Structural Topology Optimization: Theory, Practice and Software
,
Springer Science & Business Media
,
London
.
32.
Papanicolau
,
G.
,
Bensoussan
,
A.
, and
Lions
,
J.-L.
,
1978
,
Asymptotic Analysis for Periodic Structures
,
Elsevier
,
Amsterdam
.
33.
Sánchez-Palencia
,
E.
,
1980
, “
Non-Homogeneous Media and Vibration Theory
”.
Lecture Notes in Physics
,
127
.
34.
Bensoussan
,
A.
,
Lions
,
J.-L.
, and
Papanicolaou
,
G.
,
2011
,
Asymptotic Analysis for Periodic Structures
, Vol.
374
.
American Mathematical Society
.
35.
Kalamkarov
,
A. L.
,
Andrianov
,
I. V.
, and
Danishevs’kyy
,
V. V.
,
2009
, “
Asymptotic Homogenization of Composite Materials and Structures
,”
Appl. Mech. Rev.
,
62
(
3
), p.
030802
.
36.
Hill
,
R.
,
1952
, “
The Elastic Behaviour of a Crystalline Aggregate
,”
Proc. Phys. Soc. Sect. A
,
65
(
5
), p.
349
.
37.
Huang
,
B.
,
Duan
,
Y.-H.
,
Hu
,
W.-C.
,
Sun
,
Y.
, and
Chen
,
S.
,
2015
, “
Structural, Anisotropic Elastic and Thermal Properties of MB (M = Ti, Zr and Hf) Monoborides
,”
Ceram. Int.
,
41
(
5
), pp.
6831
6843
.
38.
Duan
,
Y.
,
Sun
,
Y.
,
Peng
,
M.
,
Guo
,
Z.
, and
Zhu
,
P.
,
2012
, “
Calculated Structure, Elastic and Electronic Properties of Mg2Pb At High Pressure
,”
J. Wuhan University of Technol.-Mater. Sci. Ed.
,
27
(
2
), pp.
377
381
.
39.
Peng
,
M.
,
Duan
,
Y.
, and
Sun
,
Y.
,
2015
, “
Anisotropic Elastic Properties and Electronic Structure of Sr–Pb Compounds
,”
Comput. Mater. Sci.
,
98
, pp.
311
319
.
40.
Al-Ketan
,
O.
,
Rowshan
,
R.
, and
Al-Rub
,
R. K. A.
,
2018
, “
Topology-Mechanical Property Relationship of 3D Printed Strut, Skeletal, and Sheet Based Periodic Metallic Cellular Materials
,”
Addit. Manuf.
,
19
, pp.
167
183
.
41.
Maas
,
S. A.
,
Ellis
,
B. J.
,
Ateshian
,
G. A.
, and
Weiss
,
J. A.
,
2012
, “
FEBio: Finite Elements for Biomechanics
,”
ASME J. Biomech. Eng.
,
134
(
1
), p.
011005
.
You do not currently have access to this content.