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Research Papers

Symplectic Analysis for Wrinkles: A Case Study of Layered neo-Hookean Structures

[+] Author and Article Information
Teng Zhang

Department of Mechanical and
Aerospace Engineering,
Syracuse University,
Syracuse, NY 13244
e-mail: tzhang48@syr.edu

1Corresponding author.

Manuscript received April 2, 2017; final manuscript received April 26, 2017; published online May 15, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(7), 071002 (May 15, 2017) (9 pages) Paper No: JAM-17-1177; doi: 10.1115/1.4036613 History: Received April 02, 2017; Revised April 26, 2017

Wrinkles are widely found in natural and engineering structures, ranging from skins to stretchable electronics. However, it is nontrivial to predict wrinkles, especially for complicated structures, such as multilayer or gradient structures. Here, we establish a symplectic analysis framework for the wrinkles and apply it to layered neo-Hookean structures. The symplectic structure enables us to accurately and efficiently solve the eigenvalue problems of wrinkles via the extended Wittrick–Williams (w–W) algorithm. The symplectic analysis is able to exactly predict wrinkles in bi- and triple-layer structures, compared with the benchmark results and finite element simulations. Our findings also shed light on the formation of hierarchical wrinkles

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Figures

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Fig. 1

Schematics of wrinkles in a bilayer structure: (a) flat structure at the stress-free state and (b) wrinkling under compression or film swelling

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Fig. 2

Schematics of a quad-layer structure with triple-layer films (films 1–3) adhered on a semi-infinite substitute. A pseudo-time variable t is used to describe the effective dynamics system. The state variables at each interface are also labeled.

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Fig. 3

Determine the critical strain and wavenumber of wrinkle through symplectic analysis: (a) Search the critical strain for a given wavenumber by identifying the jump of eigenvalue count, and (b) the minimum strain for all the possible wavenumber is the critical wrinkle strain εw and the associated wavenumber is the critical wrinkle wavenumber kw

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Fig. 4

Boundary value problem defined by state variables and the combination of two consecutive intervals into a larger interval

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Fig. 5

2N algorithm for calculating the eigenvalue count Jm and F,G, and Q matrices for film j

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Fig. 6

(a) Critical wrinkle wavenumber as a function of modulus mismatch ratio (film thickness is one) and (b) critical wrinkle strain as a function of modulus mismatch ratio

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Fig. 7

(a) Critical wrinkle wavenumber as a function of prestretch (film thickness is 1) and (b) critical wrinkle strain as a function of prestretch. Modulus mismatch ratio is 1000.

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Fig. 8

Symplectic analysis for wrinkles of tripe-layer structure: (a) schematic of a triple-layer structure for the modeling and analysis. Films 1 (top red) and 2 (intermediate orange) are adhered on a semi-infinite substrate (bottom blue). In the text, we also denote the films 1 and 2 as top and intermediate films to avoid confusions with other numbers, (b) relation between given wavenumber and critical strain for instability for μ2=8.37, (c) the long and short wavelengths varying with the normalized modulus of the intermediate film, and (d) wrinkle strains for the triple-layer structure predicted from symplectic analysis and FEM simulations (see color figure online).

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Fig. 9

FEM snapshots for wrinkle patterns of a triple-layer neo-Hookean structure with different μ2: (a) μ2/μs=4, (b) μ2/μs=8.37, and (c) μ2/μs=500. Scale bar is 100.

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