Originated from the art of paper cutting and folding, kirigami and origami have shown promising applications in a broad range of scientific and engineering fields. Developments of kirigami-inspired inverse design methods that map target three-dimensional (3D) geometries into two-dimensional (2D) patterns of cuts and creases are desired to serve as guidelines for practical applications. In this paper, using programed kirigami tessellations, we propose two design methods to approximate the geometries of developable surfaces and nonzero Gauss curvature surfaces with rotational symmetry. In the first method, a periodic array of kirigami pattern with spatially varying geometric parameters is obtained, allowing formation of developable surfaces of desired curvature distribution and thickness, through controlled shrinkage and bending deformations. In the second method, another type of kirigami tessellations, in combination with Miura origami, is proposed to approximate nondevelopable surfaces with rotational symmetry. Both methods are validated by experiments of folding patterned thin copper films into desired 3D structures. The mechanical behaviors of the kirigami designs are investigated using analytical modeling and finite element simulations. The proposed methods extend the design space of mechanical metamaterials and are expected to be useful for kirigami-inspired applications.

References

References
1.
Felton
,
S.
,
Tolley
,
M.
,
Demaine
,
E.
,
Rus
,
D.
, and
Wood
,
R.
,
2014
, “
A Method for Building Self-Folding Machines
,”
Science
,
345
(
6197
), pp.
644
646
.
2.
Filipov
,
E. T.
,
Tachi
,
T.
, and
Paulino
,
G. H.
,
2015
, “
Origami Tubes Assembled Into Stiff, Yet Reconfigurable Structures and Metamaterials
,”
Proc. Natl. Acad. Sci.
,
112
(
40
), pp.
12321
12326
.
3.
Lang
,
R. J.
,
1996
, “
A Computational Algorithm for Origami Design
,”
12th Annual Symposium on Computational Geometry
(
SOCG '96
), Philadelphia, PA, May 24–26, pp.
98
105
.
4.
Seffen
,
K. A.
, and
Stott
,
S. V.
,
2014
, “
Surface Texturing Through Cylinder Buckling
,”
ASME J. Appl. Mech.
,
81
(
6
), p.
061001
.
5.
Blees
,
M. K.
,
Barnard
,
A. W.
,
Rose
,
P. A.
,
Roberts
,
S. P.
,
McGill
,
K. L.
,
Huang
,
P. Y.
,
Ruyack
,
A. R.
,
Kevek
,
J. W.
,
Kobrin
,
B.
,
Muller
,
D. A.
, and
McEuen
,
P. L.
,
2015
, “
Graphene Kirigami
,”
Nature
,
524
(
7564
), pp.
204
207
.
6.
Sussman
,
D. M.
,
Cho
,
Y.
,
Castle
,
T.
,
Gong
,
X.
,
Jung
,
E.
,
Yang
,
S.
, and
Kamien
,
R. D.
,
2015
, “
Algorithmic Lattice Kirigami: A Route to Pluripotent Materials
,”
Proc. Natl. Acad. Sci.
,
112
(
24
), pp.
7449
7453
.
7.
Safsten
,
C.
,
Fillmore
,
T.
,
Logan
,
A.
,
Halverson
,
D.
, and
Howell
,
L.
,
2016
, “
Analyzing the Stability Properties of Kaleidocycles
,”
ASME J. Appl. Mech.
,
83
(
5
), p.
051001
.
8.
Silverberg
,
J. L.
,
Na
,
J.-H.
,
Evans
,
A. A.
,
Liu
,
B.
,
Hull
,
T. C.
,
Santangelo
,
C. D.
,
Lang
,
R. J.
,
Hayward
,
R. C.
, and
Cohen
,
I.
,
2015
, “
Origami Structures With a Critical Transition to Bistability Arising From Hidden Degrees of Freedom
,”
Nat. Mater.
,
14
(
4
), pp.
389
393
.
9.
Hawkes
,
E.
,
An
,
B.
,
Benbernou
,
N. M.
,
Tanaka
,
H.
,
Kim
,
S.
,
Demaine
,
E. D.
,
Rus
,
D.
, and
Wood
,
R. J.
,
2010
, “
Programmable Matter by Folding
,”
Proc. Natl. Acad. Sci.
,
107
(
28
), pp.
12441
12445
.
10.
Lamoureux
,
A.
,
Lee
,
K.
,
Shlian
,
M.
,
Forrest
,
S. R.
, and
Shtein
,
M.
,
2015
, “
Dynamic Kirigami Structures for Integrated Solar Tracking
,”
Nat. Commun.
,
6
, p.
8092
.
11.
Hanna
,
B. H.
,
Magleby
,
S. P.
,
Lang
,
R. J.
, and
Howell
,
L. L.
,
2015
, “
Force–Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms
,”
ASME J. Appl. Mech.
,
82
(
8
), p.
081001
.
12.
Overvelde
,
J. T. B.
,
Weaver
,
J. C.
,
Hoberman
,
C.
, and
Bertoldi
,
K.
,
2017
, “
Rational Design of Reconfigurable Prismatic Architected Materials
,”
Nature
,
541
(
7637
), pp.
347
352
.
13.
Yan
,
Z.
,
Zhang
,
F.
,
Wang
,
J.
,
Liu
,
F.
,
Guo
,
X.
,
Nan
,
K.
,
Lin
,
Q.
,
Gao
,
M.
,
Xiao
,
D.
,
Shi
,
Y.
,
Qiu
,
Y.
,
Luan
,
H.
,
Kim
,
J. H.
,
Wang
,
Y.
,
Luo
,
H.
,
Han
,
M.
,
Huang
,
Y.
,
Zhang
,
Y.
, and
Rogers
,
J. A.
,
2016
, “
Controlled Mechanical Buckling for Origami-Inspired Construction of 3D Microstructures in Advanced Materials
,”
Adv. Funct. Mater.
,
26
(
16
), pp.
2629
2639
.
14.
Yasuda
,
H.
, and
Yang
,
J.
,
2015
, “
Reentrant Origami-Based Metamaterials With Negative Poisson's Ratio and Bistability
,”
Phys. Rev. Lett.
,
114
(
18
), p.
185502
.
15.
Overvelde
,
J. T. B.
,
de Jong
,
T. A.
,
Shevchenko
,
Y.
,
Becerra
,
S. A.
,
Whitesides
,
G. M.
,
Weaver
,
J. C.
,
Hoberman
,
C.
, and
Bertoldi
,
K.
,
2016
, “
A Three-Dimensional Actuated Origami-Inspired Transformable Metamaterial With Multiple Degrees of Freedom
,”
Nat. Commun.
,
7
, p.
10929
.
16.
Tang
,
R.
,
Huang
,
H.
,
Tu
,
H.
,
Liang
,
H.
,
Liang
,
M.
,
Song
,
Z.
,
Xu
,
Y.
,
Jiang
,
H.
, and
Yu
,
H.
,
2014
, “
Origami-Enabled Deformable Silicon Solar Cells
,”
Appl. Phys. Lett.
,
104
(
8
), p.
083501
.
17.
Song
,
Z.
,
Ma
,
T.
,
Tang
,
R.
,
Cheng
,
Q.
,
Wang
,
X.
,
Krishnaraju
,
D.
,
Panat
,
R.
,
Chan
,
C. K.
,
Yu
,
H.
, and
Jiang
,
H.
,
2014
, “
Origami Lithium-Ion Batteries
,”
Nat. Commun.
,
5
, p.
3140
.
18.
Bassik
,
N.
,
Stern
,
G. M.
, and
Gracias
,
D. H.
,
2009
, “
Microassembly Based on Hands Free Origami With Bidirectional Curvature
,”
Appl. Phys. Lett.
,
95
(
9
), p.
091901
.
19.
Xu
,
S.
,
Yan
,
Z.
,
Jang
,
K.-I.
,
Huang
,
W.
,
Fu
,
H.
,
Kim
,
J.
,
Wei
,
Z.
,
Flavin
,
M.
,
McCracken
,
J.
,
Wang
,
R.
,
Badea
,
A.
,
Liu
,
Y.
,
Xiao
,
D.
,
Zhou
,
G.
,
Lee
,
J.
,
Chung
,
H. U.
,
Cheng
,
H.
,
Ren
,
W.
,
Banks
,
A.
,
Li
,
X.
,
Paik
,
U.
,
Nuzzo
,
R. G.
,
Huang
,
Y.
,
Zhang
,
Y.
, and
Rogers
,
J. A.
,
2015
, “
Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling
,”
Science
,
347
(
6218
), pp.
154
159
.
20.
Laflin
,
K. E.
,
Morris
,
C. J.
,
Muqeem
,
T.
, and
Gracias
,
D. H.
,
2012
, “
Laser Triggered Sequential Folding of Microstructures
,”
Appl. Phys. Lett.
,
101
(
13
), p.
131901
.
21.
Zhang
,
Y.
,
Yan
,
Z.
,
Nan
,
K.
,
Xiao
,
D.
,
Liu
,
Y.
,
Luan
,
H.
,
Fu
,
H.
,
Wang
,
X.
,
Yang
,
Q.
,
Wang
,
J.
,
Ren
,
W.
,
Si
,
H.
,
Liu
,
F.
,
Yang
,
L.
,
Li
,
H.
,
Wang
,
J.
,
Guo
,
X.
,
Luo
,
H.
,
Wang
,
L.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2015
, “
A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes
,”
Proc. Natl. Acad. Sci.
,
112
(
38
), pp.
11757
11764
.
22.
Ma
,
J.
, and
You
,
Z.
,
2014
, “
Energy Absorption of Thin-Walled Square Tubes With a Prefolded Origami Pattern—Part I: Geometry and Numerical Simulation
,”
ASME J. Appl. Mech.
,
81
(
1
), p.
011003
.
23.
Chen
,
B. G.
,
Liu
,
B.
,
Evans
,
A. A.
,
Paulose
,
J.
,
Cohen
,
I.
,
Vitelli
,
V.
, and
Santangelo
,
C. D.
,
2016
, “
Topological Mechanics of Origami and Kirigami
,”
Phys. Rev. Lett.
,
116
(
13
), p.
135501
.
24.
Chen
,
Y.
,
Peng
,
R.
, and
You
,
Z.
,
2015
, “
Origami of Thick Panels
,”
Science
,
349
(
6246
), pp.
396
400
.
25.
Sareh
,
S.
, and
Rossiter
,
J.
,
2013
, “
Kirigami Artificial Muscles With Complex Biologically Inspired Morphologies
,”
Smart Mater. Struct.
,
22
(
1
), p.
014004
.
26.
Hanna
,
B. H.
,
Lund
,
J. M.
,
Lang
,
R. J.
,
Magleby
,
S. P.
, and
Howell
,
L. L.
,
2014
, “
Waterbomb Base: A Symmetric Single-Vertex Bistable Origami Mechanism
,”
Smart Mater. Struct.
,
23
(
9
), p.
094009
.
27.
Schenk
,
M.
, and
Guest
,
S. D.
,
2013
, “
Geometry of Miura-Folded Metamaterials
,”
Proc. Natl. Acad. Sci.
,
110
(
9
), pp.
3276
3281
.
28.
Wei
,
Z. Y.
,
Guo
,
Z. V.
,
Dudte
,
L.
,
Liang
,
H. Y.
, and
Mahadevan
,
L.
,
2013
, “
Geometric Mechanics of Periodic Pleated Origami
,”
Phys. Rev. Lett.
,
110
(
21
), p.
215501
.
29.
Florijn
,
B.
,
Coulais
,
C.
, and
van Hecke
,
M.
,
2014
, “
Programmable Mechanical Metamaterials
,”
Phys. Rev. Lett.
,
113
(
17
), p.
175503
.
30.
Eidini
,
M.
, and
Paulino
,
G. H.
,
2015
, “
Unraveling Metamaterial Properties in Zigzag-Base Folded Sheets
,”
Sci. Adv.
,
1
(
8
), p.
e1500224
.
31.
Brunck
,
V.
,
Lechenault
,
F.
,
Reid
,
A.
, and
Adda-Bedia
,
M.
,
2016
, “
Elastic Theory of Origami-Based Metamaterials
,”
Phys. Rev. E
,
93
(
3
), p.
033005
.
32.
Zhou
,
X.
,
Wang
,
H.
, and
You
,
Z.
,
2015
, “
Design of Three-Dimensional Origami Structures Based on a Vertex Approach
,”
Proc. R. Soc. Math. Phys. Eng. Sci.
,
471
(
2181
), p.
20150407
.
33.
Dudte
,
L. H.
,
Vouga
,
E.
,
Tachi
,
T.
, and
Mahadevan
,
L.
,
2016
, “
Programming Curvature Using Origami Tessellations
,”
Nat. Mater.
,
15
(
5
), pp.
583
588
.
34.
Tachi
,
T.
,
2013
, “
Designing Freeform Origami Tessellations by Generalizing Resch's Patterns
,”
ASME J. Mech. Des.
,
135
(
11
), p.
111006
.
35.
Tachi
,
T.
,
2010
, “
Freeform Variations of Origami
,”
J. Geom. Graph.
,
14
(
2
), pp.
203
215
.
36.
Castle
,
T.
,
Sussman
,
D. M.
,
Tanis
,
M.
, and
Kamien
,
R. D.
,
2016
, “
Additive Lattice Kirigami
,”
Sci. Adv.
,
2
(
9
), p.
e1601258
.
37.
Nojima
,
T.
, and
Saito
,
K.
,
2006
, “
Development of Newly Designed Ultralight Core Structures
,”
Int. J. Ser. Solid Mech. Mater. Eng.
,
49
(
1
), pp.
38
42
.
38.
Shyu
,
T. C.
,
Damasceno
,
P. F.
,
Dodd
,
P. M.
,
Lamoureux
,
A.
,
Xu
,
L.
,
Shlian
,
M.
,
Shtein
,
M.
,
Glotzer
,
S. C.
, and
Kotov
,
N. A.
,
2015
, “
A Kirigami Approach to Engineering Elasticity in Nanocomposites Through Patterned Defects
,”
Nat. Mater.
,
14
(
8
), pp.
785
789
.
39.
Qi
,
Z.
,
Campbell
,
D. K.
, and
Park
,
H. S.
,
2014
, “
Atomistic Simulations of Tension-Induced Large Deformation and Stretchability in Graphene Kirigami
,”
Phys. Rev. B
,
90
(
24
), p.
245437
.
40.
Castle
,
T.
,
Cho
,
Y.
,
Gong
,
X.
,
Jung
,
E.
,
Sussman
,
D. M.
,
Yang
,
S.
, and
Kamien
,
R. D.
,
2014
, “
Making the Cut: Lattice Kirigami Rules
,”
Phys. Rev. Lett.
,
113
(
24
), p.
245502
.
41.
Cho
,
Y.
,
Shin
,
J.-H.
,
Costa
,
A.
,
Kim
,
T. A.
,
Kunin
,
V.
,
Li
,
J.
,
Lee
,
S. Y.
,
Yang
,
S.
,
Han
,
H. N.
,
Choi
,
I.-S.
, and
Srolovitz
,
D. J.
,
2014
, “
Engineering the Shape and Structure of Materials by Fractal Cut
,”
Proc. Natl. Acad. Sci.
,
111
(
49
), pp.
17390
17395
.
42.
Xie
,
R.
,
Chen
,
Y.
, and
Gattas
,
J. M.
,
2015
, “
Parametrisation and Application of Cube and Eggbox-Type Folded Geometries
,”
Int. J. Space Struct.
,
30
(
2
), pp.
99
110
.
43.
Lv
,
C.
,
Krishnaraju
,
D.
,
Konjevod
,
G.
,
Yu
,
H.
, and
Jiang
,
H.
,
2014
, “
Origami Based Mechanical Metamaterials
,”
Sci. Rep.
,
4
, p.
5979
.
44.
Schenk
,
M.
, and
Guest
,
S. D.
,
2011
, “
Origami Folding: A Structural Engineering Approach
,”
Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education
,
CRC Press
, Boca Raton, FL, pp.
291
303
.
45.
Silverberg
,
J. L.
,
Evans
,
A. A.
,
McLeod
,
L.
,
Hayward
,
R. C.
,
Hull
,
T.
,
Santangelo
,
C. D.
, and
Cohen
,
I.
,
2014
, “
Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials
,”
Science
,
345
(
6197
), pp.
647
650
.
46.
Kim
,
J.
,
Hanna
,
J. A.
,
Byun
,
M.
,
Santangelo
,
C. D.
, and
Hayward
,
R. C.
,
2012
, “
Designing Responsive Buckled Surfaces by Halftone Gel Lithography
,”
Science
,
335
(
6073
), pp.
1201
1205
.
47.
Guo
,
X.
,
Li
,
H.
,
Ahn
,
B. Y.
,
Duoss
,
E. B.
,
Hsia
,
K. J.
,
Lewis
,
J. A.
, and
Nuzzo
,
R. G.
,
2009
, “
Two-and Three-Dimensional Folding of Thin Film Single-Crystalline Silicon for Photovoltaic Power Applications
,”
Proc. Natl. Acad. Sci.
,
106
(
48
), pp.
20149
20154
.
48.
Wang
,
F.
,
Gong
,
H.
,
Chen
,
X.
, and
Chen
,
C. Q.
,
2016
, “
Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-Based Cylindrical Structures
,”
Sci. Rep.
,
6
(
1
), p.
33312
.
49.
Gattas
,
J. M.
,
Wu
,
W.
, and
You
,
Z.
,
2013
, “
Miura-Base Rigid Origami: Parameterizations of First-Level Derivative and Piecewise Geometries
,”
ASME J. Mech. Des.
,
135
(
11
), p.
111011
.
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