Abstract
Serpentine structures are of growing interest due to its unique mechanical and physical properties for applications in stretchable electronics, mechanical sensing, and biomedical devices. Mechanics-guided, deterministic three-dimensional (3D) assembly provides routes to form remarkable 3D structures, which in turn significantly improve its potential for applications. Therefore, an accurate postbuckling analysis is essential to the complex 3D serpentine structures with arbitrary geometry/material parameters. Here, simple, analytical expressions are obtained for the displacement and effective rigidity of serpentine structures during postbuckling. By tuning geometry parameters, the amplitude of assembled 3D serpentine structures can span a very broad range from zero to that of a straight ribbon. The analytical model can be used in design, fabrication, and application of versatile 3D serpentine structures to ensure their compatibility with the ultra-low rigidity biological tissues. A hierarchical 3D serpentine structure with ultra-low rigidity is presented to demonstrate the application of the analytical model.
Issue Section:
Research Papers
References
1.
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
. 2.
Zhang
, Y.
, Xu
, S.
, Fu
, H.
, 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
. 3.
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
. 4.
Pan
, T. S.
, Pharr
, M.
, Ma
, Y. J.
, Ning
, R.
, Yan
, Z.
, Xu
, R. X.
, Feng
, X.
, Huang
, Y. G.
, and Rogers
, J. A.
, 2017
, “Experimental and Theoretical Studies of Serpentine Interconnects on Ultrathin Elastomers for Stretchable Electronics
,” Adv. Funct. Mater.
, 27
(37
), p. 1702589
. 5.
Huang
, Y.
, Mu
, Z. Z.
, Feng
, P.
, and Yuan
, J. H.
, 2019
, “Mechanics Design for Compatible Deformation of Fractal Serpentine Interconnects in High-Density Stretchable Electronics
,” ASME J. Appl. Mech.
, 86
(3
), p. 031011
. 6.
Liu
, S. Y.
, Ha
, T.
, and Lu
, N. S.
, 2019
, “Experimentally and Numerically Validated Analytical Solutions to Nonbuckling Piezoelectric Serpentine Ribbons
,” ASME J. Appl. Mech.
, 86
(5
), p. 051010
. 7.
Feng
, P.
, Yuan
, J. H.
, Huang
, Y.
, and Li
, X. Y.
, 2020
, “Analytical Solutions for the Lateral-Torsional Buckling of Serpentine Interconnects in Stretchable Electronics
,” ASME J. Appl. Mech.
, 87
(8
), p. 081005
. 8.
Zhou
, Y. D.
, and Fei
, Q. G.
, 2020
, “Evaluating Deformation Modes of Sandwich Serpentine Structures for High Stretchability
,” Thin-Walled Struct.
, 157
, p. 107087
. 9.
Liu
, Y.
, Wang
, X.
, Xu
, Y.
, Xue
, Z.
, Zhang
, Y.
, Ning
, X.
, Cheng
, X.
, et al, 2019
, “Harnessing the Interface Mechanics of Hard Films and Soft Substrates for 3D Assembly by Controlled Buckling
,” Proc. Natl. Acad. Sci. U. S. A.
, 116
(31
), pp. 15368
–15377
. 10.
Li
, S. P.
, Han
, M. D.
, Rogers
, J. A.
, Zhang
, Y. H.
, Huang
, Y. G.
, and Wang
, H. L.
, 2019
, “Mechanics of Buckled Serpentine Structures Formed via Mechanics-Guided, Deterministic Three-Dimensional Assembly
,” J. Mech. Phys. Solids
, 125
, pp. 736
–748
. 11.
Nan
, K.
, Kang
, S. D.
, Li
, K.
, Yu
, K. J.
, Zhu
, F.
, Wang
, J.
, Dunn
, A. C.
, et al, 2018
, “Compliant and Stretchable Thermoelectric Coils for Energy Harvesting in Miniature Flexible Devices
,” Sci. Adv.
, 4
(11
), p. eaau5849
. 12.
Su
, Y.
, Ping
, X.
, Yu
, K. J.
, Lee
, J. W.
, Fan
, J. A.
, Wang
, B.
, Li
, M.
, et al, 2017
, “In-Plane Deformation Mechanics for Highly Stretchable Electronics
,” Adv. Mater.
, 29
(8
), p. 1604989
. 13.
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
. 14.
Jang
, K. I.
, Li
, K.
, Chung
, H. U.
, Xu
, S.
, Jung
, H. N.
, Yang
, Y.
, Kwak
, J. W.
, et al, 2017
, “Self-assembled Three Dimensional Network Designs for Soft Electronics
,” Nat. Commun.
, 8
, p. 15894
. 15.
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
. 16.
Xu
, S.
, Yan
, Z.
, Jang
, K.-I.
, Huang
, W.
, Fu
, H.
, Kim
, J.
, Wei
, Z.
, et al, 2015
, “Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling
,” Science
, 347
(6218
), pp. 154
–159
. 17.
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
. 18.
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
. 19.
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
. 20.
Zhao
, J.
, Li
, W.
, Guo
, X.
, Wang
, H.
, Rogers
, J. A.
, and Huang
, Y.
, 2020
, “Theoretical Modeling of Tunable Vibrations of Three-Dimensional Serpentine Structures for Simultaneous Measurement of Adherent Cell Mass and Modulus
,” MRS Bull.
, 46
(1
), pp. 107
–114
.21.
Han
, M.
, Wang
, H.
, Yang
, Y.
, Liang
, C.
, Bai
, W.
, Yan
, Z.
, Li
, H.
, et al, 2019
, “Three-dimensional Piezoelectric Polymer Microsystems for Vibrational Energy Harvesting, Robotic Interfaces and Biomedical Implants
,” Nat. Electron.
, 2
(1
), pp. 26
–35
. 22.
Lu
, N. S.
, Lu
, C.
, Yang
, S. X.
, and Rogers
, J.
, 2012
, “Highly Sensitive Skin-Mountable Strain Gauges Based Entirely on Elastomers
,” Adv. Funct. Mater.
, 22
(19
), pp. 4044
–4050
. 23.
Yu
, X.
, Xie
, Z.
, Yu
, Y.
, Lee
, J.
, Vazquez-Guardado
, A.
, Luan
, H.
, Ruban
, J.
, et al, 2019
, “Skin-integrated Wireless Haptic Interfaces for Virtual and Augmented Reality
,” Nature
, 575
(7783
), pp. 473
–479
. 24.
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
. 25.
Wang
, X. J.
, Feiner
, R.
, Luan
, H. W.
, Zhang
, Q. H.
, Zhao
, S. W.
, Zhang
, Y.
, Han
, M. D.
, et al, 2020
, “Three-dimensional Electronic Scaffolds for Monitoring and Regulation of Multifunctional Hybrid Tissues
,” Extreme Mech. Lett.
, 35
, p. 100634
. 26.
Kim
, D.-H.
, Lu
, N.
, Ghaffari
, R.
, Kim
, Y.-S.
, Lee
, S. P.
, Xu
, L.
, Wu
, J.
, et al, 2011
, “Materials for Multifunctional Balloon Catheters with Capabilities in Cardiac Electrophysiological Mapping and Ablation Therapy
,” Nat. Mater.
, 10
(4
), pp. 316
–323
. 27.
Zhao
, J.
, Zhang
, F.
, Guo
, X.
, Huang
, Y.
, Zhang
, Y.
, and Wang
, H.
, 2021
, “Torsional Deformation Dominated Buckling of Serpentine Structures to Form Three-Dimensional Architectures with Ultra-low Rigidity
,” J. Mech. Phys. Solids
, 155
, p. 104568
. 28.
Yan
, Z.
, Wang
, B.
, Wang
, K.
, Zhao
, S.
, Li
, S.
, Huang
, Y.
, and Wang
, H.
, 2020
, “Cellular Substrate to Facilitate Global Buckling of Serpentine Structures
,” ASME J. Appl. Mech.
, 87
(2
), p. 024501
. 29.
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.
, 2005
, “Transfer Printing by Kinetic Control of Adhesion to an Elastomeric Stamp
,” Nat. Mater.
, 5
(1
), pp. 33
–38
. 30.
DelRio
, F. W.
, de Boer
, M. P.
, Knapp
, J. A.
, David Reedy
, E.
, Clews
, P. J.
, and Dunn
, M. L.
, 2005
, “The Role of van der Waals Forces in Adhesion of Micromachined Surfaces
,” Nat. Mater.
, 4
(8
), pp. 629
–634
. 31.
Yan
, Z.
, Han
, M.
, Shi
, Y.
, Badea
, A.
, Yang
, Y.
, Kulkarni
, A.
, Hanson
, E.
, et al, 2017
, “Three-dimensional Mesostructures as High-Temperature Growth Templates, Electronic Cellular Scaffolds, and Self-Propelled Microrobots
,” Proc. Natl. Acad. Sci. U. S. A.
, 114
(45
), pp. E9455
–E9464
. 32.
Zhang
, F.
, Li
, S.
, Shen
, Z.
, Cheng
, X.
, Xue
, Z.
, Zhang
, H.
, Song
, H.
, et al, 2021
, “Rapidly Deployable and Morphable 3D Mesostructures with Applications in Multimodal Biomedical Devices
,” Proc. Natl. Acad. Sci. U. S. A.
, 118
(11
), p. e2026414118
. 33.
Yang
, Y.
, Wu
, M.
, Vazquez-Guardado
, A.
, Wegener
, A. J.
, Grajales-Reyes
, J. G.
, Deng
, Y.
, Wang
, T.
, et al, 2021
, “Wireless Multilateral Devices for Optogenetic Studies of Individual and Social Behaviors
,” Nat. Neurosci.
, 24
(7
), pp. 1035
–1045
. 34.
Xu
, Z.
, Fan
, Z.
, Zi
, Y.
, Zhang
, Y.
, and Huang
, Y.
, 2020
, “An Inverse Design Method of Buckling-Guided Assembly for Ribbon-Type 3D Structures
,” ASME J. Appl. Mech.
, 87
(3
), p. 031004
. 35.
Li
, K.
, Chen
, L.
, Zhu
, F.
, and Huang
, Y.
, 2021
, “Thermal and Mechanical Analyses of Compliant Thermoelectric Coils for Flexible and Bio-Integrated Devices
,” ASME J. Appl. Mech.
, 88
(2
), p. 021011
. 36.
Young
, W. C.
, Budynas
, R. G.
, and Sadegh
, A. M.
, 2012
, Roark's Formulas for Stress and Strain
, McGraw-Hill
, New York
.37.
Su
, Y. W.
, Wu
, J.
, Fan
, Z. C.
, Hwang
, K. C.
, Song
, J. Z.
, Huang
, Y. G.
, and Rogers
, J. A.
, 2012
, “Postbuckling Analysis and its Application to Stretchable Electronics
,” J. Mech. Phys. Solids
, 60
(3
), pp. 487
–508
. 38.
Chung
, H. U.
, Kim
, B. H.
, Lee
, J. Y.
, Lee
, J.
, Xie
, Z.
, Ibler
, E. M.
, Lee
, K.
, et al, 2019
, “Binodal, Wireless Epidermal Electronic Systems with in-Sensor Analytics for Neonatal Intensive Care
,” Science
, 363
(6430
), p. eaau0780
. Copyright © 2022 by ASME
You do not currently have access to this content.
