The ribbons selectively bonded to a prestrained elastomeric substrate may buckle into three-dimensional (3D) microstructures after the prestrain release, leading to three possible deformation modes, global, local, and no buckling, depending on the adhesion between the ribbons and substrate. This note establishes analytically the critical length-to-thickness ratio of ribbons, above which the global buckling mode (preferred for mechanically guided 3D deterministic assembly) occurs without material failure.

References

1.
Zhang
,
Y. H.
,
Zhang
,
F.
,
Yan
,
Z.
,
Ma
,
Q.
,
Li
,
X. L.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2017
, “
Printing, Folding and Assembly Methods for Forming 3D Mesostructures in Advanced Materials
,”
Nat. Rev. Mater.
,
2
(
4
), p.
17019
.
2.
Yan
,
Z.
,
Han
,
M. D.
,
Yang
,
Y. Y.
,
Nan
,
K. W.
,
Luan
,
H. W.
,
Luo
,
Y. Y.
,
Zhang
,
Y. H.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2017
, “
Deterministic Assembly of 3D Mesostructures in Advanced Materials Via Compressive Buckling: A Short Review of Recent Progress
,”
Extreme Mech. Lett.
,
11
, pp.
96
104
.
3.
Lee
,
Y. K.
,
Jang
,
K. I.
,
Ma
,
Y. J.
,
Koh
,
A.
,
Chen
,
H.
,
Jung
,
H. N.
,
Kim
,
Y.
,
Kwak
,
J. W.
,
Wang
,
L.
,
Xue
,
Y. G.
,
Yang
,
Y. Y.
,
Tian
,
W. L.
,
Jiang
,
Y.
,
Zhang
,
Y. H.
,
Feng
,
X.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2017
, “
Chemical Sensing Systems That Utilize Soft Electronics on Thin Elastomeric Substrates With Open Cellular Designs
,”
Adv. Funct. Mater.
,
27
(
9
), p.
1605476
.
4.
Wu
,
J. T.
,
Yuan
,
C.
,
Ding
,
Z.
,
Isakov
,
M.
,
Mao
,
Y. Q.
,
Wang
,
T. J.
,
Dunn
,
M. L.
, and
Qi
,
H. J.
,
2016
, “
Multi-Shape Active Composites by 3D Printing of Digital Shape Memory Polymers
,”
Sci. Rep.
,
6
(
1
), p.
24224
.
5.
Rus
,
D.
, and
Tolley
,
M. T.
,
2015
, “
Design, Fabrication and Control of Soft Robots
,”
Nature
,
521
(
7553
), pp.
467
475
.
6.
Lee
,
S.
,
Kim
,
S.
,
Kim
,
T. T.
,
Kim
,
Y.
,
Choi
,
M.
,
Lee
,
S. H.
,
Kim
,
J. Y.
, and
Min
,
B.
,
2012
, “
Reversibly Stretchable and Tunable Terahertz Metamaterials With Wrinkled Layouts
,”
Adv. Mater.
,
24
(
26
), pp.
3491
3497
.
7.
Soukoulis
,
C. M.
, and
Wegener
,
M.
,
2011
, “
Past Achievements and Future Challenges in the Development of Three-Dimensional Photonic Metamaterials
,”
Nature Photon.
,
5
(
9
), pp.
523
530
.
8.
Ma
,
Q.
, and
Zhang
,
Y. H.
,
2016
, “
Mechanics of Fractal-Inspired Horseshoe Microstructures for Applications in Stretchable Electronics
,”
ASME J. Appl. Mech.
,
83
(
11
), p.
111008
.
9.
Adams
,
J. J.
,
Duoss
,
E. B.
,
Malkowski
,
T. F.
,
Motala
,
M. J.
,
Ahn
,
B. Y.
,
Nuzzo
,
R. G.
,
Bernhard
,
J. T.
, and
Lewis
,
J. A.
,
2011
, “
Conformal Printing of Electrically Small Antennas on Three-Dimensional Surfaces
,”
Adv. Mater.
,
23
(
11
), pp.
1335
1340
.
10.
Skylar-Scott
,
M. A.
,
Gunasekaran
,
S.
, and
Lewis
,
J. A.
,
2016
, “
Laser-Assisted Direct Ink Writing of Planar and 3D Metal Architectures
,”
Proc. Natl. Acad. Sci. U.S.A.
,
113
(
22
), pp.
6137
6142
.
11.
Cumpston
,
B. H.
,
Ananthavel
,
S. P.
,
Barlow
,
S.
,
Dyer
,
D. L.
,
Ehrlich
,
J. E.
,
Erskine
,
L. L.
,
Heikal
,
A. A.
,
Kuebler
,
S. M.
,
Lee
,
I. Y. S.
,
McCord-Maughon
,
D.
,
Qin
,
J. Q.
,
Rockel
,
H.
,
Rumi
,
M.
,
Wu
,
X. L.
,
Marder
,
S. R.
, and
Perry
,
J. W.
,
1999
, “
Two-Photon Polymerization Initiators for Three-Dimensional Optical Data Storage and Microfabrication
,”
Nature
,
398
(
6722
), pp.
51
54
.
12.
Kawata
,
S.
,
Sun
,
H. B.
,
Tanaka
,
T.
, and
Takada
,
K.
,
2001
, “
Finer Features for Functional Microdevices—Micromachines Can Be Created With Higher Resolution Using Two-Photon Absorption
,”
Nature
,
412
(
6848
), pp.
697
698
.
13.
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
.
14.
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
.
15.
Wang
,
F.
,
Guo
,
X. G.
,
Xu
,
J. X.
,
Zhang
,
Y. H.
, and
Chen
,
C. Q.
,
2017
, “
Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs
,”
ASME J. Appl. Mech.
,
84
(
6
), p.
061007
.
16.
Gladman
,
A. S.
,
Matsumoto
,
E. A.
,
Nuzzo
,
R. G.
,
Mahadevan
,
L.
, and
Lewis
,
J. A.
,
2016
, “
Biomimetic 4D Printing
,”
Nat. Mater.
,
15
(
4
), pp.
413
418
.
17.
Ma
,
Y. J.
,
Feng
,
X.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y. H.
,
2017
, “
Design and Application of ‘J-Shaped' Stress–Strain Behavior in Stretchable Electronics: A Review
,”
Lab Chip
,
17
(
10
), pp.
1689
1704
.
18.
Xu
,
S.
,
Yan
,
Z.
,
Jang
,
K. I.
,
Huang
,
W.
,
Fu
,
H. R.
,
Kim
,
J.
,
Wei
,
Z.
,
Flavin
,
M.
,
McCracken
,
J.
,
Wang
,
R.
,
Badea
,
A.
,
Liu
,
Y.
,
Xiao
,
D. Q.
,
Zhou
,
G. Y.
,
Lee
,
J.
,
Chung
,
H. U.
,
Cheng
,
H. Y.
,
Ren
,
W.
,
Banks
,
A.
,
Li
,
X. L.
,
Paik
,
U.
,
Nuzzo
,
R. G.
,
Huang
,
Y. G.
,
Zhang
,
Y. H.
, and
Rogers
,
J. A.
,
2015
, “
Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling
,”
Science
,
347
(
6218
), pp.
154
159
.
19.
Huang
,
Y.
,
Yuan
,
J. H.
,
Zhang
,
Y. C.
, and
Feng
,
X.
,
2016
, “
Interfacial Delamination of Inorganic Films on Viscoelastic Substrates
,”
ASME J. Appl. Mech.
,
83
(
10
), p.
101005
.
20.
Ma
,
Y. J.
,
Jang
,
K. I.
,
Wang
,
L.
,
Jung
,
H. N.
,
Kwak
,
J. W.
,
Xue
,
Y. G.
,
Chen
,
H.
,
Yang
,
Y. Y.
,
Shi
,
D. W.
,
Feng
,
X.
,
Rogers
,
J. A.
, and
Huang
,
Y. G.
,
2016
, “
Design of Strain-Limiting Substrate Materials for Stretchable and Flexible Electronics
,”
Adv. Funct. Mater.
,
26
(
29
), pp.
5345
5351
.
21.
Raayai-Ardakani
,
S.
,
Yague
,
J. L.
,
Gleason
,
K. K.
, and
Boyce
,
M. C.
,
2016
, “
Mechanics of Graded Wrinkling
,”
ASME J. Appl. Mech.
,
83
(
12
), p.
121011
.
22.
Ko
,
H. C.
,
Shin
,
G.
,
Wang
,
S. D.
,
Stoykovich
,
M. P.
,
Lee
,
J. W.
,
Kim
,
D. H.
,
Ha
,
J. S.
,
Huang
,
Y. G.
,
Hwang
,
K. C.
, and
Rogers
,
J. A.
,
2009
, “
Curvilinear Electronics Formed Using Silicon Membrane Circuits and Elastomeric Transfer Elements
,”
Small
,
5
(
23
), pp.
2703
2709
.
23.
Timoshenko
,
S. P.
, and
Gere
,
J. M.
,
1961
,
Theory of Elastic Stability
,
McGraw-Hill
, New York.
24.
Wang
,
S. D.
,
Xiao
,
J. L.
,
Song
,
J. Z.
,
Ko
,
H. C.
,
Hwang
,
K. C.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2010
, “
Mechanics of Curvilinear Electronics
,”
Soft Matter
,
6
(
22
), pp.
5757
5763
.
25.
Khang
,
D. Y.
,
Jiang
,
H. Q.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2006
, “
A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates
,”
Science
,
311
(
5758
), pp.
208
212
.
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