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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.

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Figures

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. 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. 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. 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. 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.

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