Fractal-inspired designs represent an emerging class of strategy for stretchable electronics, which have been demonstrated to be particularly useful for various applications, such as stretchable batteries and biointegrated electrophysiological electrodes. The fractal-inspired constructs usually undergo complicated, nonlinear deformations under mechanical loading, because of the highly complex and diverse microstructures inherent in high-order fractal patterns. The underlying relations between the nonlinear mechanical responses and microstructure geometry are essential in practical applications, which require a relevant mechanics theory to serve as the basis of a design approach. Here, a theoretical model inspired by the mechanism of ordered unraveling is developed to study the nonlinear stress–strain curves and elastic stretchability for a class of fractal-inspired horseshoe microstructures. Analytic solutions were obtained for some key mechanical quantities, such as the elastic modulus and the tangent modulus at the beginning of each deformation stage. Both the finite-element analyses (FEA) and experiments were carried out to validate the model. Systematic analyses of the microstructure–property relationship dictate how to leverage the various geometric parameters to tune the multistage, J-shaped stress–strain curves. Moreover, a demonstrative example shows the utility of the theoretical model in design optimization of fractal-inspired microstructures used as electrophysiological electrodes, aiming to achieve maximum elastic stretchability for prescribed filling ratios. The results indicate a substantial enhancement (e.g., >4 times) of elastic stretchability by using fractal designs, as compared to traditional horseshoe designs. This study can serve as design guidelines of fractal-inspired microstructures in different stretchable electronic systems.

References

1.
Pang
,
C.
,
Lee
,
C.
, and
Suh
,
K. Y.
,
2013
, “
Recent Advances in Flexible Sensors for Wearable and Implantable Devices
,”
J. Appl. Polym. Sci.
,
130
(
3
), pp.
1429
1441
.
2.
Abe
,
Y.
,
Chinzei
,
T.
,
Mabuchi
,
K.
,
Snyder
,
A. J.
,
Isoyama
,
T.
,
Imanishi
,
K.
,
Yonezawa
,
T.
,
Matsuura
,
H.
,
Kouno
,
A.
,
Ono
,
T.
,
Atsumi
,
K.
,
Fujimasa
,
I.
, and
Imachi
,
K.
,
1998
, “
Physiological Control of a Total Artificial Heart: Conductance- and Arterial Pressure-Based Control
,”
J. Appl. Physiol.
,
84
(
3
), pp.
868
876
.http://jap.physiology.org/content/84/3/868.short
3.
Rogers
,
J. A.
,
Someya
,
T.
, and
Huang
,
Y. G.
,
2010
, “
Materials and Mechanics for Stretchable Electronics
,”
Science
,
327
(
5973
), pp.
1603
1607
.
4.
Vanfleteren
,
J.
,
Gonzalez
,
M.
,
Bossuyt
,
F.
,
Hsu
,
Y. Y.
,
Vervust
,
T.
,
De Wolf
,
I.
, and
Jablonski
,
M.
,
2012
, “
Printed Circuit Board Technology Inspired Stretchable Circuits
,”
MRS Bull.
,
37
(
3
), pp.
254
260
.
5.
Jung
,
S.
,
Kim
,
J. H.
,
Kim
,
J.
,
Choi
,
S.
,
Lee
,
J.
,
Park
,
I.
,
Hyeon
,
T.
, and
Kim
,
D. H.
,
2014
, “
Reverse-Micelle-Induced Porous Pressure-Sensitive Rubber for Wearable Human-Machine Interfaces
,”
Adv. Mater.
,
26
(
28
), pp.
4825
4830
.
6.
Lanzara
,
G.
,
Salowitz
,
N.
,
Guo
,
Z. Q.
, and
Chang
,
F. K.
,
2010
, “
A Spider-Web-Like Highly Expandable Sensor Network for Multifunctional Materials
,”
Adv. Mater.
,
22
(
41
), pp.
4643
4648
.
7.
Kim
,
D. H.
,
Lu
,
N. S.
,
Ma
,
R.
,
Kim
,
Y. S.
,
Kim
,
R. H.
,
Wang
,
S. D.
,
Wu
,
J.
,
Won
,
S. M.
,
Tao
,
H.
,
Islam
,
A.
,
Yu
,
K. J.
,
Kim
,
T. I.
,
Chowdhury
,
R.
,
Ying
,
M.
,
Xu
,
L. Z.
,
Li
,
M.
,
Chung
,
H. J.
,
Keum
,
H.
,
McCormick
,
M.
,
Liu
,
P.
,
Zhang
,
Y. W.
,
Omenetto
,
F. G.
,
Huang
,
Y. G.
,
Coleman
,
T.
, and
Rogers
,
J. A.
,
2011
, “
Epidermal Electronics
,”
Science
,
333
(
6044
), pp.
838
843
.
8.
Yao
,
S. S.
, and
Zhu
,
Y.
,
2014
, “
Wearable Multifunctional Sensors Using Printed Stretchable Conductors Made of Silver Nanowires
,”
Nanoscale
,
6
(
4
), pp.
2345
2352
.
9.
Someya
,
T.
,
Sekitani
,
T.
,
Iba
,
S.
,
Kato
,
Y.
,
Kawaguchi
,
H.
, and
Sakurai
,
T.
,
2004
, “
A Large-Area, Flexible Pressure Sensor Matrix With Organic Field-Effect Transistors for Artificial Skin Applications
,”
Proc. Natl. Acad. Sci. U. S. A.
,
101
(
27
), pp.
9966
9970
.
10.
Wagner
,
S.
,
Lacour
,
S. P.
,
Jones
,
J.
,
Hsu
,
P. H. I.
,
Sturm
,
J. C.
,
Li
,
T.
, and
Suo
,
Z. G.
,
2004
, “
Electronic Skin: Architecture and Components
,”
Phys. E: Low-Dimens. Syst. Nanostruct.
,
25
(
2–3
), pp.
326
334
.
11.
Mannsfeld
,
S. C. B.
,
Tee
,
B. C. K.
,
Stoltenberg
,
R. M.
,
Chen
,
C.
,
Barman
,
S.
,
Muir
,
B. V. O.
,
Sokolov
,
A. N.
,
Reese
,
C.
, and
Bao
,
Z. N.
,
2010
, “
Highly Sensitive Flexible Pressure Sensors With Microstructured Rubber Dielectric Layers
,”
Nat. Mater.
,
9
(
10
), pp.
859
864
.
12.
Kim
,
J.
,
Lee
,
M.
,
Shim
,
H. J.
,
Ghaffari
,
R.
,
Cho
,
H. R.
,
Son
,
D.
,
Jung
,
Y. H.
,
Soh
,
M.
,
Choi
,
C.
,
Jung
,
S.
,
Chu
,
K.
,
Jeon
,
D.
,
Lee
,
S. T.
,
Kim
,
J. H.
,
Choi
,
S. H.
,
Hyeon
,
T.
, and
Kim
,
D. H.
,
2014
, “
Stretchable Silicon Nanoribbon Electronics for Skin Prosthesis
,”
Nat. Commun.
,
5
, p.
5747
.
13.
Yang
,
S. X.
,
Chen
,
Y. C.
,
Nicolini
,
L.
,
Pasupathy
,
P.
,
Sacks
,
J.
,
Su
,
B.
,
Yang
,
R.
,
Sanchez
,
D.
,
Chang
,
Y. F.
,
Wang
,
P. L.
,
Schnyer
,
D.
,
Neikirk
,
D.
, and
Lu
,
N. S.
,
2015
, “
‘Cut-and-Paste’ Manufacture of Multiparametric Epidermal Sensor Systems
,”
Adv. Mater.
,
27
(
41
), pp.
6423
6430
.
14.
Yu
,
Z.
,
Graudejus
,
O.
,
Tsay
,
C.
,
Lacour
,
S. P.
,
Wagner
,
S.
, and
Morrison
,
B.
,
2009
, “
Monitoring Hippocampus Electrical Activity In Vitro on an Elastically Deformable Microelectrode Array
,”
J. Neurotrauma
,
26
(
7
), pp.
1135
1145
.
15.
Kim
,
D. H.
,
Lu
,
N. S.
,
Ghaffari
,
R.
,
Kim
,
Y. S.
,
Lee
,
S. P.
,
Xu
,
L. Z.
,
Wu
,
J. A.
,
Kim
,
R. H.
,
Song
,
J. Z.
,
Liu
,
Z. J.
,
Viventi
,
J.
,
de Graff
,
B.
,
Elolampi
,
B.
,
Mansour
,
M.
,
Slepian
,
M. J.
,
Hwang
,
S.
,
Moss
,
J. D.
,
Won
,
S. M.
,
Huang
,
Y. G.
,
Litt
,
B.
, and
Rogers
,
J. A.
,
2011
, “
Materials for Multifunctional Balloon Catheters With Capabilities in Cardiac Electrophysiological Mapping and Ablation Therapy
,”
Nat. Mater.
,
10
(
4
), pp.
316
323
.
16.
Lee
,
H.
,
Choi
,
T. K.
,
Lee
,
Y. B.
,
Cho
,
H. R.
,
Ghaffari
,
R.
,
Wang
,
L.
,
Choi
,
H. J.
,
Chung
,
T. D.
,
Lu
,
N. S.
,
Hyeon
,
T.
,
Choi
,
S. H.
, and
Kim
,
D. H.
,
2016
, “
A Graphene-Based Electrochemical Device With Thermoresponsive Microneedles for Diabetes Monitoring and Therapy
,”
Nat. Nanotechnol.
,
11
(
6
), pp.
566
572
.
17.
Zhang
,
Y. H.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2015
, “
Mechanics of Stretchable Batteries and Supercapacitors
,”
Curr. Opin. Solid State Mater. Sci.
,
19
(
3
), pp.
190
199
.
18.
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
.
19.
Kim
,
D. H.
,
Ahn
,
J. H.
,
Choi
,
W. M.
,
Kim
,
H. S.
,
Kim
,
T. H.
,
Song
,
J. Z.
,
Huang
,
Y. G. Y.
,
Liu
,
Z. J.
,
Lu
,
C.
, and
Rogers
,
J. A.
,
2008
, “
Stretchable and Foldable Silicon Integrated Circuits
,”
Science
,
320
(
5875
), pp.
507
511
.
20.
Yu
,
C. J.
,
Masarapu
,
C.
,
Rong
,
J. P.
,
Wei
,
B. Q.
, and
Jiang
,
H. Q.
,
2009
, “
Stretchable Supercapacitors Based on Buckled Single-Walled Carbon Nanotube Macrofilms
,”
Adv. Mater.
,
21
(
47
), pp.
4793
4797
.
21.
Zang
,
J. F.
,
Ryu
,
S.
,
Pugno
,
N.
,
Wang
,
Q. M.
,
Tu
,
Q.
,
Buehler
,
M. J.
, and
Zhao
,
X. H.
,
2013
, “
Multifunctionality and Control of the Crumpling and Unfolding of Large-Area Graphene
,”
Nat. Mater.
,
12
(
4
), pp.
321
325
.
22.
Zang
,
J. F.
,
Cao
,
C. Y.
,
Feng
,
Y. Y.
,
Liu
,
J.
, and
Zhao
,
X. H.
,
2014
, “
Stretchable and High-Performance Supercapacitors With Crumpled Graphene Papers
,”
Sci. Rep.
,
4
, p.
6492
.
23.
Jiang
,
H. Q.
,
Khang
,
D. Y.
,
Song
,
J. Z.
,
Sun
,
Y. G.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2007
, “
Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports
,”
Proc. Natl. Acad. Sci. U. S. A.
,
104
(
40
), pp.
15607
15612
.
24.
Li
,
Y. H.
,
Fang
,
B.
,
Zhang
,
J. H.
, and
Song
,
J. Z.
,
2011
, “
Surface Effects on the Wrinkling of Piezoelectric Films on Compliant Substrates
,”
J. Appl. Phys.
,
110
(
11
), p.
114303
.
25.
Zhu
,
S. Z.
, and
Li
,
T.
,
2014
, “
Wrinkling Instability of Graphene on Substrate-Supported Nanoparticles
,”
ASME J. Appl. Mech.
,
81
(
6
), p.
061008
.
26.
Wang
,
Q. M.
, and
Zhao
,
X. H.
,
2014
, “
Phase Diagrams of Instabilities in Compressed Film-Substrate Systems
,”
ASME J. Appl. Mech.
,
81
(
5
), p.
051004
.
27.
Lacour
,
S. P.
,
Jones
,
J.
,
Wagner
,
S.
,
Li
,
T.
, and
Suo
,
Z. G.
,
2005
, “
Stretchable Interconnects for Elastic Electronic Surfaces
,”
Proc. IEEE
,
93
(
8
), pp.
1459
1467
.
28.
Ko
,
H. C.
,
Stoykovich
,
M. P.
,
Song
,
J. Z.
,
Malyarchuk
,
V.
,
Choi
,
W. M.
,
Yu
,
C. J.
,
Geddes
,
J. B.
,
Xiao
,
J. L.
,
Wang
,
S. D.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2008
, “
A Hemispherical Electronic Eye Camera Based on Compressible Silicon Optoelectronics
,”
Nature
,
454
(
7205
), pp.
748
753
.
29.
Kim
,
D. H.
,
Song
,
J. Z.
,
Choi
,
W. M.
,
Kim
,
H. S.
,
Kim
,
R. H.
,
Liu
,
Z. J.
,
Huang
,
Y. Y.
,
Hwang
,
K. C.
,
Zhang
,
Y. W.
, and
Rogers
,
J. A.
,
2008
, “
Materials and Noncoplanar Mesh Designs for Integrated Circuits With Linear Elastic Responses to Extreme Mechanical Deformations
,”
Proc. Natl. Acad. Sci. U. S. A.
,
105
(
48
), pp.
18675
18680
.
30.
Zhang
,
Y. H.
,
Wang
,
S. D.
,
Li
,
X. T.
,
Fan
,
J. A.
,
Xu
,
S.
,
Song
,
Y. M.
,
Choi
,
K. J.
,
Yeo
,
W. H.
,
Lee
,
W.
,
Nazaar
,
S. N.
,
Lu
,
B. W.
,
Yin
,
L.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2014
, “
Experimental and Theoretical Studies of Serpentine Microstructures Bonded To Prestrained Elastomers for Stretchable Electronics
,”
Adv. Funct. Mater.
,
24
(
14
), pp.
2028
2037
.
31.
Li
,
T.
,
Suo
,
Z. G.
,
Lacour
,
S. P.
, and
Wagner
,
S.
,
2005
, “
Compliant Thin Film Patterns of Stiff Materials as Platforms for Stretchable Electronics
,”
J. Mater. Res.
,
20
(
12
), pp.
3274
3277
.
32.
Gonzalez
,
M.
,
Axisa
,
F.
,
Buicke
,
M. V.
,
Brosteaux
,
D.
,
Vandevelde
,
B.
, and
Vanfleteren
,
J.
,
2008
, “
Design of Metal Interconnects for Stretchable Electronic Circuits
,”
Microelectron. Reliab.
,
48
(
6
), pp.
825
832
.
33.
Huang
,
Y. A.
,
Wang
,
Y. Z.
,
Xiao
,
L.
,
Liu
,
H. M.
,
Dong
,
W. T.
, and
Yin
,
Z. P.
,
2014
, “
Microfluidic Serpentine Antennas With Designed Mechanical Tunability
,”
Lab Chip
,
14
(
21
), pp.
4205
4212
.
34.
Widlund
,
T.
,
Yang
,
S. X.
,
Hsu
,
Y. Y.
, and
Lu
,
N. S.
,
2014
, “
Stretchability and Compliance of Freestanding Serpentine-Shaped Ribbons
,”
Int. J. Solids Struct.
,
51
(
23–24
), pp.
4026
4037
.
35.
Yang
,
S. X.
,
Su
,
B.
,
Bitar
,
G.
, and
Lu
,
N. S.
,
2014
, “
Stretchability of Indium Tin Oxide (ITO) Serpentine Thin Films Supported by Kapton Substrates
,”
Int. J. Fract.
,
190
(
1–2
), pp.
99
110
.
36.
Shi
,
X. T.
,
Xu
,
R. X.
,
Li
,
Y. H.
,
Zhang
,
Y. H.
,
Ren
,
Z. G.
,
Gu
,
J. F.
,
Rogers
,
J. A.
, and
Huang
,
Y. G.
,
2014
, “
Mechanics Design for Stretchable, High Areal Coverage GaAs Solar Module on an Ultrathin Substrate
,”
ASME J. Appl. Mech.
,
81
(
12
), p.
124502
.
37.
Lv
,
C.
,
Yu
,
H.
, and
Jiang
,
H.
,
2014
, “
Archimedean Spiral Design for Extremely Stretchable Interconnects
,”
Extreme Mech. Lett.
,
1
, pp.
29
34
.
38.
Xu
,
S.
,
Zhang
,
Y. H.
,
Cho
,
J.
,
Lee
,
J.
,
Huang
,
X.
,
Jia
,
L.
,
Fan
,
J. A.
,
Su
,
Y. W.
,
Su
,
J.
,
Zhang
,
H. G.
,
Cheng
,
H. Y.
,
Lu
,
B. W.
,
Yu
,
C. J.
,
Chuang
,
C.
,
Kim
,
T. I.
,
Song
,
T.
,
Shigeta
,
K.
,
Kang
,
S.
,
Dagdeviren
,
C.
,
Petrov
,
I.
,
Braun
,
P. V.
,
Huang
,
Y.
,
Paik
,
U.
, and
Rogers
,
J. A.
,
2013
, “
Stretchable Batteries With Self-Similar Serpentine Interconnects and Integrated Wireless Recharging Systems
,”
Nat. Commun.
,
4
, p.
1543
.
39.
Fan
,
J. A.
,
Yeo
,
W. H.
,
Su
,
Y. W.
,
Hattori
,
Y.
,
Lee
,
W.
,
Jung
,
S. Y.
,
Zhang
,
Y. H.
,
Liu
,
Z. J.
,
Cheng
,
H. Y.
,
Falgout
,
L.
,
Bajema
,
M.
,
Coleman
,
T.
,
Gregoire
,
D.
,
Larsen
,
R. J.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2014
, “
Fractal Design Concepts for Stretchable Electronics
,”
Nat. Commun.
,
5
, p.
3266
.
40.
Xu
,
S.
,
Zhang
,
Y. H.
,
Jia
,
L.
,
Mathewson
,
K. E.
,
Jang
,
K. I.
,
Kim
,
J.
,
Fu
,
H. R.
,
Huang
,
X.
,
Chava
,
P.
,
Wang
,
R. H.
,
Bhole
,
S.
,
Wang
,
L. Z.
,
Na
,
Y. J.
,
Guan
,
Y.
,
Flavin
,
M.
,
Han
,
Z. S.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2014
, “
Soft Microfluidic Assemblies of Sensors, Circuits, and Radios for the Skin
,”
Science
,
344
(
6179
), pp.
70
74
.
41.
Huang
,
Y. A.
,
Dong
,
W. T.
,
Huang
,
T.
,
Wang
,
Y. Z.
,
Xiao
,
L.
,
Su
,
Y. W.
, and
Yin
,
Z. P.
,
2015
, “
Self-Similar Design for Stretchable Wireless LC Strain Sensors
,”
Sens. Actuators A
,
224
, pp.
36
42
.
42.
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
.
43.
Song
,
Z. M.
,
Ma
,
T.
,
Tang
,
R.
,
Cheng
,
Q.
,
Wang
,
X.
,
Krishnaraju
,
D.
,
Panat
,
R.
,
Chan
,
C. K.
,
Yu
,
H. Y.
, and
Jiang
,
H. Q.
,
2014
, “
Origami Lithium-Ion Batteries
,”
Nat. Commun.
,
5
, p.
3140
.
44.
Song
,
Z. M.
,
Wang
,
X.
,
Lv
,
C.
,
An
,
Y. H.
,
Liang
,
M. B.
,
Ma
,
T.
,
He
,
D.
,
Zheng
,
Y. J.
,
Huang
,
S. Q.
,
Yu
,
H. Y.
, and
Jiang
,
H. Q.
,
2015
, “
Kirigami-Based Stretchable Lithium-Ion Batteries
,”
Sci. Rep.
,
5
, p.
10988
.
45.
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. U. S. A.
,
111
(
49
), pp.
17390
17395
.
46.
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
.
47.
Zhang
,
Y. H.
,
Yan
,
Z.
,
Nan
,
K. W.
,
Xiao
,
D. Q.
,
Liu
,
Y. H.
,
Luan
,
H. W.
,
Fu
,
H. R.
,
Wang
,
X. Z.
,
Yang
,
Q. L.
,
Wang
,
J. C.
,
Ren
,
W.
,
Si
,
H. Z.
,
Liu
,
F.
,
Yang
,
L. H.
,
Li
,
H. J.
,
Wang
,
J. T.
,
Guo
,
X. L.
,
Luo
,
H. Y.
,
Wang
,
L.
,
Huang
,
Y. G.
, and
Rogers
,
J. A.
,
2015
, “
A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes
,”
Proc. Natl. Acad. Sci. U. S. A.
,
112
(
38
), pp.
11757
11764
.
48.
Zhang
,
Y. H.
,
Fu
,
H. R.
,
Xu
,
S.
,
Fan
,
J. A.
,
Hwang
,
K. C.
,
Jiang
,
J. Q.
,
Rogers
,
J. A.
, and
Huang
,
Y. G.
,
2014
, “
A Hierarchical Computational Model for Stretchable Interconnects With Fractal-Inspired Designs
,”
J. Mech. Phys. Solids
,
72
, pp.
115
130
.
49.
Zhang
,
Y. H.
,
Fu
,
H. R.
,
Su
,
Y. W.
,
Xu
,
S.
,
Cheng
,
H. Y.
,
Fan
,
J. A.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2013
, “
Mechanics of Ultra-Stretchable Self-Similar Serpentine Interconnects
,”
Acta Mater.
,
61
(
20
), pp.
7816
7827
.
50.
Su
,
Y. W.
,
Wang
,
S. D.
,
Huang
,
Y. A.
,
Luan
,
H. W.
,
Dong
,
W. T.
,
Fan
,
J. A.
,
Yang
,
Q. L.
,
Rogers
,
J. A.
, and
Huang
,
Y. G.
,
2015
, “
Elasticity of Fractal Inspired Interconnects
,”
Small
,
11
(
3
), pp.
367
373
.
51.
Zhang
,
Y. H.
,
Xu
,
S.
,
Fu
,
H. R.
,
Lee
,
J.
,
Su
,
J.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2013
, “
Buckling in Serpentine Microstructures and Applications in Elastomer-Supported Ultra-Stretchable Electronics With High Areal Coverage
,”
Soft Matter
,
9
(
33
), pp.
8062
8070
.
52.
Ma
,
Q.
,
Cheng
,
H. Y.
,
Jang
,
K. I.
,
Luan
,
H. W.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
,
Huang
,
Y. G.
, and
Zhang
,
Y. H.
,
2016
, “
A Nonlinear Mechanics Model of Bio-Inspired Hierarchical Lattice Materials Consisting of Horseshoe Microstructures
,”
J. Mech. Phys. Solids
,
90
, pp.
179
202
.
53.
Meitl
,
M. A.
,
Zhu
,
Z. T.
,
Kumar
,
V.
,
Lee
,
K. J.
,
Feng
,
X.
,
Huang
,
Y. Y.
,
Adesida
,
I.
,
Nuzzo
,
R. G.
, and
Rogers
,
J. A.
,
2006
, “
Transfer Printing by Kinetic Control of Adhesion to an Elastomeric Stamp
,”
Nat. Mater.
,
5
(
1
), pp.
33
38
.
54.
Carlson
,
A.
,
Bowen
,
A. M.
,
Huang
,
Y. G.
,
Nuzzo
,
R. G.
, and
Rogers
,
J. A.
,
2012
, “
Transfer Printing Techniques for Materials Assembly and Micro/Nanodevice Fabrication
,”
Adv. Mater.
,
24
(
39
), pp.
5284
5318
.
You do not currently have access to this content.