Abstract

Stretchable variable-curvature interconnects, such as two-dimensional spirals, offer certain advantages over widely used serpentine and horseshoe designs with constant curvatures in terms of stretchability and coverage. A clear understanding of the nonlinear stretching mechanics of variable-curvature interconnects is essential for its optimization and application. This work develops a unified mechanics model based on finite deformation theory for general interconnects defined by parametric curves. Theoretical predictions for stretched configurations, effective tensile stress, and maximum strain are validated by finite element analysis. The effects of geometric shape parameters on the mechanical responses and stretchability of sinusoidal-serpentine and sinusoidal-spiral interconnects are thoroughly investigated. This work provides certain insights for designing stretchable, planar interconnects for advanced flexible electronics.

References

1.
Chortos
,
A.
,
Liu
,
J.
, and
Bao
,
Z. N.
,
2016
, “
Pursuing Prosthetic Electronic Skin
,”
Nat. Mater.
,
15
(
9
), pp.
937
950
.
2.
Kim
,
D.-H.
,
Lu
,
N.
,
Ma
,
R.
,
Kim
,
Y.-S.
,
Kim
,
R.-H.
,
Wang
,
S.
,
Wu
,
J.
, et al
,
2011
, “
Epidermal Electronics
,”
Science
,
333
(
6044
), pp.
838
843
.
3.
Ko
,
H. C.
,
Stoykovich
,
M. P.
,
Song
,
J.
,
Malyarchuk
,
V.
,
Choi
,
W. M.
,
Yu
,
C.-J.
,
Geddes
,
J. B.
, III
, et al
,
2008
, “
A Hemispherical Electronic Eye Camera Based on Compressible Silicon Optoelectronics
,”
Nature
,
454
(
7205
), pp.
748
753
.
4.
Rao
,
Z.
,
Lu
,
Y.
,
Li
,
Z.
,
Sim
,
K.
,
Ma
,
Z.
,
Xiao
,
J.
, and
Yu
,
C.
,
2021
, “
Curvy, Shape-Adaptive Imagers Based on Printed Optoelectronic Pixels With a Kirigami Design
,”
Nat. Electron.
,
4
(
7
), pp.
513
521
.
5.
Brooks
,
A. K.
,
Chakravarty
,
S.
,
Ali
,
M.
, and
Yadavalli
,
V. K.
,
2022
, “
Kirigami-Inspired Biodesign for Applications in Healthcare
,”
Adv. Mater.
,
34
(
18
), p.
2109550
.
6.
Li
,
S.
,
Lu
,
D.
,
Li
,
S.
,
Liu
,
J.
,
Xu
,
Y.
,
Yan
,
Y.
,
Rodriguez
,
J. Z.
, et al
,
2024
, “
Bioresorbable, Wireless, Passive Sensors for Continuous pH Measurements and Early Detection of Gastric Leakage
,”
Sci. Adv.
,
10
(
16
), p.
eadj0268
.
7.
Han
,
M.
,
Chen
,
L.
,
Aras
,
K.
,
Liang
,
C.
,
Chen
,
X.
,
Zhao
,
H.
,
Li
,
K.
, et al
,
2020
, “
Catheter-Integrated Soft Multilayer Electronic Arrays for Multiplexed Sensing and Actuation During Cardiac Surgery
,”
Nat. Biomed. Eng.
,
4
(
10
), pp.
997
1009
.
8.
Gil
,
B.
,
Li
,
B.
,
Gao
,
A.
, and
Yang
,
G.-Z.
,
2020
, “
Miniaturized Piezo Force Sensor for a Medical Catheter and Implantable Device
,”
ACS Appl. Electron. Mater.
,
2
(
8
), pp.
2669
2677
.
9.
Khang
,
D.-Y.
,
Jiang
,
H.
,
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
.
10.
Cheng
,
H.
,
Zhang
,
Y.
,
Hwang
,
K.-C.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2014
, “
Buckling of a Stiff Thin Film on a Pre-strained Bi-layer Substrate
,”
Int. J. Solids Struct.
,
51
(
18
), pp.
3113
3118
.
11.
Yuan
,
X.
,
Won
,
S. M.
,
Han
,
M.
,
Wang
,
Y.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Wang
,
H.
,
2021
, “
Mechanics of Encapsulated Three-Dimensional Structures for Simultaneous Sensing of Pressure and Shear Stress
,”
J. Mech. Phys. Solids
,
151
, p.
104400
.
12.
Yuan
,
X.
,
Zhao
,
P.
,
Fan
,
Q.
,
Wang
,
Y.
, and
Li
,
X.
,
2024
, “
Theoretical and Numerical Analysis on Buckling Instability in a Thin Film Sandwiched Between Two Finite-Thickness Substrates Under In-plane Compression
,”
Int. J. Solids Struct.
,
304
, p.
113037
.
13.
Li
,
K.
,
Shuai
,
Y.
,
Cheng
,
X.
,
Luan
,
H.
,
Liu
,
S.
,
Yang
,
C.
,
Xue
,
Z.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2022
, “
Island Effect in Stretchable Inorganic Electronics
,”
Small
,
18
(
17
), p.
2107879
.
14.
Xu
,
S.
,
Zhang
,
Y.
,
Cho
,
J.
,
Lee
,
J.
,
Huang
,
X.
,
Jia
,
L.
,
Fan
,
J. A.
, et al
,
2013
, “
Stretchable Batteries With Self-similar Serpentine Interconnects and Integrated Wireless Recharging Systems
,”
Nat. Commun.
,
4
(
1
), p.
1543
.
15.
Cao
,
Y.
,
Zhang
,
G.
,
Zhang
,
Y.
,
Yue
,
M.
,
Chen
,
Y.
,
Cai
,
S.
,
Xie
,
T.
, and
Feng
,
X.
,
2018
, “
Direct Fabrication of Stretchable Electronics on a Polymer Substrate With Process-Integrated Programmable Rigidity
,”
Adv. Funct. Mater.
,
28
(
50
), p.
1804604
.
16.
Cai
,
M.
,
Nie
,
S.
,
Du
,
Y.
,
Wang
,
C.
, and
Song
,
J.
,
2019
, “
Soft Elastomers With Programmable Stiffness as Strain-Isolating Substrates for Stretchable Electronics
,”
ACS Appl. Mater. Interfaces
,
11
(
15
), pp.
14340
14346
.
17.
Li
,
H.
,
Wang
,
Z.
,
Lu
,
S.
,
Ma
,
Y.
, and
Feng
,
X.
,
2019
, “
Elastomers With Microislands as Strain Isolating Substrates for Stretchable Electronics
,”
Adv. Mater. Technol.
,
4
(
2
), p.
1800365
.
18.
Jin
,
L.
, and
Yang
,
S.
,
2023
, “
Engineering Kirigami Frameworks Toward Real-World Applications
,”
Adv. Mater.
,
36
(
9
) p.
2308560
.
19.
Jang
,
B.
,
Won
,
S.
,
Kim
,
J.
,
Kim
,
J.
,
Oh
,
M.
,
Lee
,
H.-J.
, and
Kim
,
J.-H.
,
2022
, “
Auxetic Meta-Display: Stretchable Display Without Image Distortion
,”
Adv. Funct. Mater.
,
32
(
22
), p.
2113299
.
20.
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
(
1
), p.
3140
.
21.
Zhu
,
Y.
,
Schenk
,
M.
, and
Filipov
,
E. T.
,
2022
, “
A Review on Origami Simulations: From Kinematics, to Mechanics, Toward Multiphysics
,”
ASME Appl. Mech. Rev.
,
74
(
3
), p.
030801
.
22.
Lu
,
L.
,
Leanza
,
S.
, and
Zhao
,
R. R.
,
2023
, “
Origami With Rotational Symmetry: A Review on Their Mechanics and Design
,”
ASME Appl. Mech. Rev.
,
75
(
5
), p.
050801
.
23.
Shi
,
X.
,
Xu
,
R.
,
Li
,
Y.
,
Zhang
,
Y.
,
Ren
,
Z.
,
Gu
,
J.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2014
, “
Mechanics Design for Stretchable, High Areal Coverage GaAs Solar Module on an Ultrathin Substrate
,”
ASME J. Appl. Mech.
,
81
(
12
), p.
124502
.
24.
Oh
,
H.
,
Oh
,
J.-Y.
,
Park
,
C. W.
,
Pi
,
J.-E.
,
Yang
,
J.-H.
, and
Hwang
,
C.-S.
,
2022
, “
High Density Integration of Stretchable Inorganic Thin Film Transistors With Excellent Performance and Reliability
,”
Nat. Commun.
,
13
(
1
), p.
4963
.
25.
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
.
26.
Song
,
J. Z.
,
Huang
,
Y. G.
,
Xiao
,
J. L.
,
Wang
,
S. D.
,
Hwang
,
K. C.
,
Ko
,
H. C.
,
Kim
,
D.-H.
,
Stoykovich
,
M. P.
, and
Rogers
,
J. A.
,
2009
, “
Mechanics of Noncoplanar Mesh Design for Stretchable Electronic Circuits
,”
J. Appl. Phys.
,
105
(
12
), p.
123516
.
27.
Yuan
,
X.
,
Zhu
,
B.
, and
Wang
,
Y.
,
2022
, “
Post-Buckling Evolution of Compressed Thin Films Adhered to Rigid Substrates
,”
Int. J. Mech. Sci.
,
232
, p.
107616
.
28.
Fan
,
J. A.
,
Yeo
,
W.-H.
,
Su
,
Y.
,
Hattori
,
Y.
,
Lee
,
W.
,
Jung
,
S.-Y.
,
Zhang
,
Y.
, et al
,
2014
, “
Fractal Design Concepts for Stretchable Electronics
,”
Nat. Commun.
,
5
(
1
), p.
3266
.
29.
Tang
,
R.
, and
Fu
,
H.
,
2020
, “
Mechanics of Buckled Kirigami Membranes for Stretchable Interconnects in Island-Bridge Structures
,”
ASME J. Appl. Mech.
,
87
(
5
), p.
051002
.
30.
Yiming
,
B.
,
Wu
,
L.
,
Zhang
,
M.
,
Han
,
Z.
,
Zhao
,
P.
,
Li
,
T.
,
Jia
,
Z.
, and
Qu
,
S.
,
2020
, “
Highly Stretchable Bilayer Lattice Structures That Elongate Via In-plane Deformation
,”
Adv. Funct. Mater.
,
30
(
12
), p.
1909473
.
31.
Drack
,
M.
,
Graz
,
I.
,
Sekitani
,
T.
,
Someya
,
T.
,
Kaltenbrunner
,
M.
, and
Bauer
,
S.
,
2015
, “
An Imperceptible Plastic Electronic Wrap
,”
Adv. Mater.
,
27
(
1
), pp.
34
40
.
32.
Miles
,
R. W.
,
Zoppi
,
G.
, and
Forbes
,
I.
,
2007
, “
Inorganic Photovoltaic Cells
,”
Mater. Today
,
10
(
11
), pp.
20
27
.
33.
Khan
,
H. A.
,
Tawalbeh
,
M.
,
Aljawrneh
,
B.
,
Abuwatfa
,
W.
,
Al-Othman
,
A.
,
Sadeghifar
,
H.
, and
Olabi
,
A. G.
,
2024
, “
A Comprehensive Review on Supercapacitors: Their Promise to Flexibility, High Temperature, Materials, Design, and Challenges
,”
Energy
,
295
, p.
131043
.
34.
Naik
,
K. G.
,
Vishnugopi
,
B. S.
,
Datta
,
J.
,
Datta
,
D.
, and
Mukherjee
,
P. P.
,
2023
, “
Electro-Chemo-Mechanical Challenges and Perspective in Lithium Metal Batteries
,”
ASME Appl. Mech. Rev.
,
75
(
1
), p.
010802
.
35.
Li
,
R.
,
Li
,
M.
,
Su
,
Y.
,
Song
,
J.
, and
Ni
,
X.
,
2013
, “
An Analytical Mechanics Model for the Island-Bridge Structure of Stretchable Electronics
,”
Soft Matter
,
9
(
35
), pp.
8476
8482
.
36.
Zhang
,
Y.
,
Fu
,
H.
,
Su
,
Y.
,
Xu
,
S.
,
Cheng
,
H.
,
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
.
37.
Su
,
Y.
,
Wang
,
S.
,
Huang
,
Y.
,
Luan
,
H.
,
Dong
,
W.
,
Fan
,
J. A.
,
Yang
,
Q.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2015
, “
Elasticity of Fractal Inspired Interconnects
,”
Small
,
11
(
3
), pp.
367
373
.
38.
Ma
,
Q.
,
Cheng
,
H.
,
Jang
,
K.-I.
,
Luan
,
H.
,
Hwang
,
K.-C.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2016
, “
A Nonlinear Mechanics Model of Bio-inspired Hierarchical Lattice Materials Consisting of Horseshoe Microstructures
,”
J. Mech. Phys. Solids
,
90
, pp.
179
202
.
39.
Yan
,
Z. G.
,
Wang
,
B. L.
, and
Wang
,
K. F.
,
2018
, “
Thermal Effects on the Structural Response of Planar Serpentine Interconnects
,”
Int. J. Mech. Sci.
,
135
, pp.
23
30
.
40.
Fan
,
Z.
,
Zhang
,
Y.
,
Ma
,
Q.
,
Zhang
,
F.
,
Fu
,
H.
,
Hwang
,
K.-C.
, and
Huang
,
Y.
,
2016
, “
A Finite Deformation Model of Planar Serpentine Interconnects for Stretchable Electronics
,”
Int. J. Solids Struct.
,
91
, pp.
46
54
.
41.
Dong
,
W.
,
Zhu
,
C.
,
Ye
,
D.
, and
Huang
,
Y.
,
2017
, “
Optimal Design of Self-Similar Serpentine Interconnects Embedded in Stretchable Electronics
,”
Appl. Phys. A
,
123
(
6
), p.
428
.
42.
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
.
43.
Yang
,
S.
,
Qiao
,
S.
, and
Lu
,
N.
,
2016
, “
Elasticity Solutions to Nonbuckling Serpentine Ribbons
,”
ASME J. Appl. Mech.
,
84
(
2
), p.
021004
.
44.
Shi
,
Y.
,
Zhao
,
J.
,
Zhang
,
B.
,
Qin
,
J.
,
Hu
,
X.
,
Cheng
,
Y.
,
Yu
,
J.
,
Jie
,
J.
, and
Zhang
,
X.
,
2024
, “
Freestanding Serpentine Silicon Strips With Ultrahigh Stretchability Over 300% for Wearable Electronics
,”
Adv. Mater.
,
36
(
24
), p.
2313603
.
45.
Su
,
Y. W.
,
Zhao
,
H. Y.
,
Liu
,
S. Y.
,
Li
,
R.
,
Wang
,
Y. H.
,
Wang
,
Y. Z.
,
Bian
,
J.
, and
Huang
,
Y. A.
,
2019
, “
Buckling of Beams With Finite Prebuckling Deformation
,”
Int. J. Solids Struct.
,
165
, pp.
148
159
.
46.
Lv
,
C.
,
Yu
,
H.
, and
Jiang
,
H.
,
2014
, “
Archimedean Spiral Design for Extremely Stretchable Interconnects
,”
Extreme Mech. Lett.
,
1
, pp.
29
34
.
47.
Rehman
,
M. U.
, and
Rojas
,
J. P.
,
2017
, “
Optimization of Compound Serpentine–Spiral Structure for Ultra-stretchable Electronics
,”
Extreme Mech. Lett.
,
15
, pp.
44
50
.
48.
Alcheikh
,
N.
,
Shaikh
,
S. F.
, and
Hussain
,
M. M.
,
2018
, “
Ultra-stretchable Archimedean Interconnects for Stretchable Electronics
,”
Extreme Mech. Lett.
,
24
, pp.
6
13
.
49.
Liu
,
Y.
,
Yan
,
Z.
,
Lin
,
Q.
,
Guo
,
X.
,
Han
,
M.
,
Nan
,
K.
,
Hwang
,
K.-C.
,
Huang
,
Y.
,
Zhang
,
Y.
, and
Rogers
,
J. A.
,
2016
, “
Guided Formation of 3D Helical Mesostructures by Mechanical Buckling: Analytical Modeling and Experimental Validation
,”
Adv. Funct. Mater.
,
26
(
17
), pp.
2909
2918
.
50.
Chang
,
J.
,
Yan
,
D.
,
Liu
,
J.
,
Zhang
,
F.
, and
Zhang
,
Y.
,
2022
, “
Mechanics of Three-Dimensional Soft Network Materials With a Class of Bio-inspired Designs
,”
ASME J. Appl. Mech.
,
89
(
7
), p.
071004
.
51.
Zhang
,
Y.
,
Wu
,
J.
,
Ma
,
Y.
,
Chen
,
H.
,
Chen
,
Y.
,
Lu
,
B.
, and
Feng
,
X.
,
2018
, “
A Finite Deformation Theory for the Climbing Habits and Attachment of Twining Plants
,”
J. Mech. Phys. Solids
,
116
, pp.
171
184
.
52.
Yuan
,
X.
, and
Wang
,
Y.
,
2023
, “
Nonlinear Stretching Mechanics of Planar Archimedean-Spiral Interconnects for Flexible Electronics
,”
Thin-Walled Struct
.,
185
, p.
110568
.
53.
Dong
,
L.
,
Wang
,
D.
,
Wang
,
J.
,
Jiang
,
C.
,
Wang
,
H.
,
Zhang
,
B.
,
Wu
,
M. S.
, and
Gu
,
G.
,
2022
, “
Modeling and Design of Periodic Polygonal Lattices Constructed From Microstructures With Varying Curvatures
,”
Phys. Rev. Appl.
,
17
(
4
), p.
044032
.
54.
Dong
,
L.
,
Wang
,
J.
, and
Wang
,
D.
,
2023
, “
Modeling and Design of Three-Dimensional Voxel Printed Lattice Metamaterials
,”
Addit. Manuf.
,
69
, p.
103532
.
55.
Rogers
,
J. A.
,
Someya
,
T.
, and
Huang
,
Y. G.
,
2010
, “
Materials and Mechanics for Stretchable Electronics
,”
Science
,
327
(
5973
), pp.
1603
1607
.
56.
Wang
,
Y.
,
Wang
,
C.
, and
Tan
,
H.
,
2020
, “
Geometry-Dependent Stretchability and Stiffness of Ribbon Kirigami Based on Large Curvature Curved Beam Model
,”
Int. J. Solids Struct.
,
182–183
, pp.
236
253
.
57.
Ma
,
Y.
,
Feng
,
X.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2017
, “
Design and Application of ‘J-Shaped’ Stress–Strain Behavior in Stretchable Electronics: A Review
,”
Lab Chip
,
17
(
10
), pp.
1689
1704
.
58.
Yin
,
Y.
,
Zhao
,
Z.
, and
Li
,
Y.
,
2021
, “
Theoretical and Experimental Research on Anisotropic and Nonlinear Mechanics of Periodic Network Materials
,”
J. Mech. Phys. Solids
,
152
, p.
104458
.
59.
Li
,
N.
,
Zhao
,
W.
,
Li
,
F.
,
Liu
,
L.
,
Liu
,
Y.
, and
Leng
,
J.
,
2023
, “
A 4D-Printed Programmable Soft Network With Fractal Design and Adjustable Hydrophobic Performance
,”
Matter
,
6
(
3
), pp.
940
962
.
60.
Jang
,
K.-I.
,
Chung
,
H. U.
,
Xu
,
S.
,
Lee
,
C. H.
,
Luan
,
H.
,
Jeong
,
J.
,
Cheng
,
H.
, et al
,
2015
, “
Soft Network Composite Materials With Deterministic and Bio-inspired Designs
,”
Nat. Commun.
,
6
(
1
), p.
6566
.
61.
Kim
,
D.-H.
,
Ahn
,
J.-H.
,
Choi
,
W. M.
,
Kim
,
H.-S.
,
Kim
,
T.-H.
,
Song
,
J.
,
Huang
,
Y. Y.
,
Liu
,
Z.
,
Lu
,
C.
, and
Rogers
,
J. A.
,
2008
, “
Stretchable and Foldable Silicon Integrated Circuits
,”
Science
,
320
(
5875
), pp.
507
511
.
62.
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
.
63.
Liu
,
J.
,
Yan
,
D.
, and
Zhang
,
Y.
,
2021
, “
Mechanics of Unusual Soft Network Materials With Rotatable Structural Nodes
,”
J. Mech. Phys. Solids
,
146
, p.
104210
.
64.
Huang
,
Y.
,
Mu
,
Z.
,
Feng
,
P.
, and
Yuan
,
J.
,
2019
, “
Mechanics Design for Compatible Deformation of Fractal Serpentine Interconnects in High-Density Stretchable Electronics
,”
ASME J. Appl. Mech.
,
86
(
3
), p.
031011
.
You do not currently have access to this content.