Research Papers

Elasticity Solutions to Nonbuckling Serpentine Ribbons

[+] Author and Article Information
Shixuan Yang

Center for Mechanics of Solids, Structures, and Materials,
Department of Aerospace Engineering and Engineering Mechanics,
The University of Texas at Austin,
210 E 24th Street,
Austin, TX 78712

Shutao Qiao

Center for Mechanics of Solids,
Structures, and Materials,
Department of Aerospace Engineering and Engineering Mechanics,
The University of Texas at Austin,
210 E 24th Street,
Austin, TX 78712

Nanshu Lu

Center for Mechanics of Solids,
Structures, and Materials,
Department of Aerospace Engineering
and Engineering Mechanics,
The University of Texas at Austin,
210 E 24th Street,
Austin, TX 78712;
Department of Biomedical Engineering,
The University of Texas at Austin,
107 W Dean Keeton St.,
Austin, TX 78712;
Texas Materials Institute,
The University of Texas at Austin,
204 E. Dean Keeton St.,
Austin, TX 78712
e-mail: nanshulu@utexas.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received September 13, 2016; final manuscript received October 31, 2016; published online November 17, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(2), 021004 (Nov 17, 2016) (9 pages) Paper No: JAM-16-1447; doi: 10.1115/1.4035118 History: Received September 13, 2016; Revised October 31, 2016

Stretchable electronics have found wide applications in bio-mimetic and bio-integrated electronics attributing to their softness, stretchability, and conformability. Although conventional electronic materials are intrinsically stiff and brittle, silicon and metal membranes can be patterned into in-plane serpentine ribbons for enhanced stretchability and compliance. While freestanding thin serpentine ribbons may easily buckle out-of-plane, thick serpentine ribbons may remain unbuckled upon stretching. Curved beam (CB) theory has been applied to analytically solve the strain field and the stiffness of freestanding, nonbuckling serpentine ribbons. While being able to fully capture the strain and stiffness of narrow serpentines, the theory cannot provide accurate solutions to serpentine ribbons whose widths are comparable to the arc radius. Here we report elasticity solutions to accurately capture nonbuckling, wide serpentine ribbons. We have demonstrated that weak boundary conditions are sufficient for solving Airy stress functions except when the serpentine’s total curve length approaches the ribbon width. Slightly modified weak boundary conditions are proposed to resolve this difficulty. Final elasticity solutions are fully validated by finite element models (FEM) and are compared with results obtained by the curved beam theory. When the serpentine ribbons are embedded in polymer matrices, their stretchability may be compromised due to the fact that the matrix can constrain the in-plane rotation of the serpentine. Comparison between the analytical solutions for freestanding serpentines and the FEM solutions for matrix-embedded serpentines reveals that matrix constraint remains trivial until the matrix modulus approaches that of the serpentine ribbon.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Rogers, J. A. , Someya, T. , and Huang, Y. G. , 2010, “ Materials and Mechanics for Stretchable Electronics,” Science, 327(5973), pp. 1603–1607. [CrossRef] [PubMed]
Suo, Z. G. , 2012, “ Mechanics of Stretchable Electronics and Soft Machines,” MRS Bull., 37(3), pp. 218–225. [CrossRef]
Hammock, M. L. , Chortos, A. , Tee, B. C. K. , Tok, J. B. H. , and Bao, Z. A. , 2013, “ 25th Anniversary Article: The Evolution of Electronic Skin (E-Skin): A Brief History, Design Considerations, and Recent Progress,” Adv. Mater., 25(42), pp. 5997–6037. [CrossRef] [PubMed]
Song, Y. M. , Xie, Y. Z. , Malyarchuk, V. , Xiao, J. L. , Jung, I. , Choi, K. J. , Liu, Z. J. , Park, H. , Lu, C. F. , Kim, R. H. , Li, R. , Crozier, K. B. , Huang, Y. G. , and Rogers, J. A. , 2013, “ Digital Cameras With Designs Inspired by the Arthropod Eye,” Nature, 497(7447), pp. 95–99. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Kang, S. K. , Murphy, R. K. J. , Hwang, S. W. , Lee, S. M. , Harburg, D. V. , Krueger, N. A. , Shin, J. H. , Gamble, P. , Cheng, H. Y. , Yu, S. , Liu, Z. J. , McCall, J. G. , Stephen, M. , Ying, H. Z. , Kim, J. , Park, G. , Webb, R. C. , Lee, C. H. , Chung, S. J. , Wie, D. S. , Gujar, A. D. , Vemulapalli, B. , Kim, A. H. , Lee, K. M. , Cheng, J. J. , Huang, Y. G. , Lee, S. H. , Braun, P. V. , Ray, W. Z. , and Rogers, J. A. , 2016, “ Bioresorbable Silicon Electronic Sensors for the Brain,” Nature, 530(7588), pp. 71–76. [CrossRef] [PubMed]
Lipomi, D. J. , Tee, B. C. K. , Vosgueritchian, M. , and Bao, Z. N. , 2011, “ Stretchable Organic Solar Cells,” Adv. Mater, 23(15), pp. 1771–1775. [CrossRef] [PubMed]
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. G. , Paik, U. , and Rogers, J. A. , 2013, “ Stretchable Batteries With Self-Similar Serpentine Interconnects and Integrated Wireless Recharging Systems,” Nat. Commun., 4, p. 1543. [CrossRef] [PubMed]
Shield, T. W. , Kim, K. S. , and Shield, R. T. , 1994, “ The Buckling of an Elastic Layer Bonded to an Elastic Substrate in Plane-Strain,” ASME J. Appl. Mech., 61(2), pp. 231–235. [CrossRef]
Bowden, N. , Brittain, S. , Evans, A. G. , Hutchinson, J. W. , and Whitesides, G. M. , 1998, “ Spontaneous Formation of Ordered Structures in Thin Films of Metals Supported on an Elastomeric Polymer,” Nature, 393(6681), pp. 146–149. [CrossRef]
Huang, R. , and Suo, Z. , 2002, “ Instability of a Compressed Elastic Film on a Viscous Layer,” Int. J. Solids Struct., 39(7), pp. 1791–1802. [CrossRef]
Lacour, S. P. , Wagner, S. , Huang, Z. Y. , and Suo, Z. , 2003, “ Stretchable Gold Conductors on Elastomeric Substrates,” Appl. Phys. Lett., 82(15), pp. 2404–2406. [CrossRef]
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. [CrossRef] [PubMed]
Qi, Y. , Kim, J. , Nguyen, T. D. , Lisko, B. , Purohit, P. K. , and McAlpine, M. C. , 2011, “ Enhanced Piezoelectricity and Stretchability in Energy Harvesting Devices Fabricated From Buckled PZT Ribbons,” Nano Lett., 11(3), pp. 1331–1336. [CrossRef] [PubMed]
Wang, Y. , Yang, R. , Shi, Z. W. , Zhang, L. C. , Shi, D. X. , Wang, E. , and Zhang, G. Y. , 2011, “ Super-Elastic Graphene Ripples for Flexible Strain Sensors,” ACS Nano, 5(5), pp. 3645–3650. [CrossRef] [PubMed]
Gray, D. S. , Tien, J. , and Chen, C. S. , 2004, “ High-Conductivity Elastomeric Electronics,” Adv. Mater., 16(5), pp. 393–397. [CrossRef]
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. [CrossRef]
Brosteaux, D. , Axisa, F. , Gonzalez, M. , and Vanfleteren, J. , 2007, “ Design and Fabrication of Elastic Interconnections for Stretchable Electronic Circuits,” IEEE Electron. Device Lett., 28(7), pp. 552–554. [CrossRef]
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. [CrossRef] [PubMed]
Hsu, Y. Y. , Gonzalez, M. , Bossuyt, F. , Axisa, F. , Vanfleteren, J. , and De Wolf, I. , 2009, “ In Situ Observations on Deformation Behavior and Stretching-Induced Failure of Fine Pitch Stretchable Interconnect,” J. Mater. Res., 24(12), pp. 3573–3582. [CrossRef]
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. [CrossRef] [PubMed]
Kim, R. H. , Bae, M. H. , Kim, D. G. , Cheng, H. Y. , Kim, B. H. , Kim, D. H. , Li, M. , Wu, J. , Du, F. , Kim, H. S. , Kim, S. , Estrada, D. , Hong, S. W. , Huang, Y. G. , Pop, E. , and Rogers, J. A. , 2011, “ Stretchable, Transparent Graphene Interconnects for Arrays of Microscale Inorganic Light Emitting Diodes on Rubber Substrates,” Nano Lett., 11(9), pp. 3881–3886. [CrossRef] [PubMed]
Ma, T. , Wang, Y. , Tang, R. , Yu, H. , and Jiang, H. , 2013, “ Pre-Patterned ZnO Nanoribbons on Soft Substrates for Stretchable Energy Harvesting Applications,” J. Appl. Phys., 113(20), p. 204503. [CrossRef]
Yang, S. , Ng, E. , and Lu, N. , 2015, “ Indium Tin Oxide (ITO) Serpentine Ribbons on Soft Substrates Stretched Beyond 100%,” Extreme Mech. Lett., 2, pp. 37–45. [CrossRef]
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. [CrossRef]
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. [PubMed]
Zhang, Y. , Fu, H. , Xu, S. , Fan, J. A. , Hwang, K.-C. , Jiang, J. , Rogers, J. A. , and Huang, Y. , 2014, “ A Hierarchical Computational Model for Stretchable Interconnects With Fractal-Inspired Designs,” J. Mech. Phys. Solids, 72, pp. 115–130. [CrossRef]
Son, D. , Lee, J. , Lee, D. J. , Ghaffari, R. , Yun, S. , Kim, S. J. , Lee, J. E. , Cho, H. R. , Yoon, S. , Yang, S. X. , Lee, S. , Qiao, S. T. , Ling, D. S. , Shin, S. , Song, J. K. , Kim, J. , Kim, T. , Lee, H. , Kim, J. , Soh, M. , Lee, N. , Hwang, C. S. , Nam, S. , Lu, N. S. , Hyeon, T. , Choi, S. H. , and Kim, D. H. , 2015, “ Bioresorbable Electronic Stent Integrated With Therapeutic Nanoparticles for Endovascular Diseases,” ACS Nano, 9(6), pp. 5937–5946. [CrossRef] [PubMed]
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. [CrossRef] [PubMed]
Liu, L. , and Lu, N. , 2016, “ Variational Formulations, Instabilities and Critical Loadings of Space Curved Beams,” Int. J. Solids Struct., 87, pp. 48–60. [CrossRef]
Zhang, Y. H. , Xu, S. , Fu, H. R. , Lee, J. , Su, J. , Hwang, K. C. , Rogers, J. A. , and Huang, Y. G. , 2013, “ Buckling in Serpentine Microstructures and Applications in Elastomer-Supported Ultra-Stretchable Electronics With High Areal Coverage,” Soft Matter, 9(33), pp. 8062–8070. [CrossRef] [PubMed]
Fan, Z. C. , Zhang, Y. H. , Ma, Q. , Zhang, F. , Fu, H. R. , Hwang, K. C. , and Huang, Y. G. , 2016, “ A Finite Deformation Model of Planar Serpentine Interconnects for Stretchable Electronics,” Int. J. Solids Struct., 91, pp. 46–54. [CrossRef] [PubMed]
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. [CrossRef]
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. [CrossRef] [PubMed]
Hsu, Y. Y. , Gonzalez, M. , Bossuyt, F. , Axisa, F. , Vanfleteren, J. , and DeWolf, I. , 2010, “ The Effect of Pitch on Deformation Behavior and the Stretching-Induced Failure of a Polymer-Encapsulated Stretchable Circuit,” J. Micromech. Microeng., 20(7), p. 075036. [CrossRef]
Hsu, Y. Y. , Gonzalez, M. , Bossuyt, F. , Axisa, F. , Vanfleteren, J. , and De Wolf, I. , 2011, “ The Effects of Encapsulation on Deformation Behavior and Failure Mechanisms of Stretchable Interconnects,” Thin Solid Films, 519(7), pp. 2225–2234. [CrossRef]
Hsu, Y. Y. , Gonzalez, M. , Bossuyt, F. , Vanfleteren, J. , and De Wolf, I. , 2011, “ Polyimide-Enhanced Stretchable Interconnects: Design, Fabrication, and Characterization,” IEEE Trans. Electron. Devices, 58(8), pp. 2680–2688. [CrossRef]
Yang, S. , Su, B. , Bitar, G. , and Lu, N. , 2014, “ Stretchability of Indium Tin Oxide (ITO) Serpentine Thin Films Supported by Kapton Substrates,” Int. J. Fracture, 190(1–2), pp. 99–110. [CrossRef]
Xu, R. , Jang, K.-I. , Ma, Y. , Jung, H. N. , Yang, Y. , Cho, M. , Zhang, Y. , Huang, Y. , and Rogers, J. A. , 2014, “ Fabric-Based Stretchable Electronics With Mechanically Optimized Designs and Prestrained Composite Substrates,” Extreme Mech. Lett., 1, pp. 120–126. [CrossRef]
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. [CrossRef]
Bandodkar, A. J. , Jeerapan, I. , You, J. M. , Nunez-Flores, R. , and Wang, J. , 2016, “ Highly Stretchable Fully-Printed CNT-Based Electrochemical Sensors and Biofuel Cells: Combining Intrinsic and Design-Induced Stretchability,” Nano Lett., 16(1), pp. 721–727. [CrossRef] [PubMed]
Yang, S. , Chen, Y. C. , Nicolini, L. , Pasupathy, P. , Sacks, J. , Becky, S. , Yang, R. , Daniel, S. , Chang, Y. F. , Wang, P. , Schnyer, D. , Neikirk, D. , and Lu, N. , 2015, “ “Cut-and-Paste” Manufacture of Multiparametric Epidermal Sensor Systems,” Adv. Mater., 27(41), pp. 6423–6430. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

(a) Three-dimensional schematic of the unit cell of a freestanding periodic serpentine ribbon with geometric parameters and boundary conditions labeled. (b) Simplified plane strain boundary value problem (BVP) of a nonbuckling serpentine ribbon. (c) The three-dimensional design space for serpentine shapes defined by three dimensionless geometric parameters.

Grahic Jump Location
Fig. 4

Comparison of strain field obtained by elasticity theory (left frames) and FEM (right frames) for serpentines with geometric parameters (a) w/R=1, l/R=0,α=0, (b) w/R=1,l/R=0,α=−50 deg, and (c) w/R=1, l/R=0,α=50 deg

Grahic Jump Location
Fig. 3

The normalized offsets when α=−π/2: (a) c0/w and (b) d0/w analytically solved as functions of w/R. Difference in εmax/εapp with and without the offsets for (c) narrow serpentines (w/R=0.2) and (d) wide serpentines (w/R=1).

Grahic Jump Location
Fig. 2

Boundary conditions for three decomposed sub-BVPs: (a) and (b) are two sub-BVPs for the arc and (c) is for the arm. (d) Definition of the c offset. (e) Definition of the d offset. (f) Illustration of the local and global coordinate systems.

Grahic Jump Location
Fig. 5

Normalized maximum strain obtained by elasticity theory (solid curve), CB theory (dashed curve), and FEM (dot) when (a) and (b) α=0, (c) w/R=0.2, and (d) w/R=1. (e) Arc angle at peak strain αp (left axis) and value of peak strain (εmax/εapp)p (right axis) plotted as functions of w/R. (f) Critical arc angle αc above which εmax/εapp<1 plotted as a function of w/R for various l/R.

Grahic Jump Location
Fig. 6

Normalized effective stiffness of serpentines PS/(2E¯wu0) plotted for (a) narrow serpentines (w/R=0.2) and (b) wide serpentines (w/R=1)

Grahic Jump Location
Fig. 7

(a) Boundary conditions for plane strain freestanding and polymer embedded serpentines. (b) εmax/εapp as a function of the matrix modulus Ematrix: dots are FEM results of embedded serpentines whereas solid and dashed lines are elasticity and CB solutions, respectively, of freestanding serpentines.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In