On the wrinkling of a pre-stressed annular thin film in tension

Coman, Ciprian D.; Bassom, Andrew P.

doi:10.1016/j.jmps.2007.01.006

Cited by 55 publications

(84 citation statements)

References 17 publications

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“…Interestingly, this result coincides with the upper bound implied by the positivity of [15] and also ensures continuity of the hoop stress at r ¼ L FFT . Thus, energy minimization naturally yields a value for the stress at the tip of the wrinkles that smoothly matches the flat region in the film to the wrinkled one.…”

Section: Far-from-threshold Wrinklingsupporting

confidence: 81%

“…15 by using numerical and asymptotic analysis (15), but to our knowledge the above force balance argument has not been noted before. ‡…”

Section: Near-threshold Wrinklingmentioning

confidence: 86%

“…Here the threshold line was obtained from a linear analysis similar to ref. 15 (albeit with a free edge BC at r ¼ R in to eliminate a spurious boundary layer near r ¼ R in ).…”

Section: Discussionmentioning

confidence: 99%

“…Whereas the energy and wrinkled extent in the NT regime are determined by Eqs. 6 and 7, bifurcation analysis [assuming f ðrÞ is infinitesimal † ] yields the value of τ c ðϵÞ and the number of wrinkles m c ðϵÞ of the emerging pattern (15,16).…”

Section: Near-threshold Wrinklingmentioning

confidence: 99%

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Prototypical model for tensional wrinkling in thin sheets

Davidovitch

Schroll

Vella

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

215

306

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The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length-a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.pattern formation | thin-film buckling T hin films are among the ubiquitous examples of flexible structures that buckle under compressive loads. More interestingly, these buckling instabilities usually develop into wrinkled patterns that provide a dramatic display of the applied stress field (1, 2). Wrinkles align perpendicularly to the compression direction, depicting the principal lines of stress and providing a geometric tool for mechanical characterization. Traditional buckling theory is regularly used to understand these patterns in the near-threshold (NT) regime, in which the deformations are small perturbations of the initial flat state. However, it has been known since Wagner (3, 4) that, when the exerted loads are well in excess of those necessary to initiate buckling, the asymptotic state of the plate is very different from the one observed under NT conditions. In this far-from-threshold (FFT) regime, the stress nearly vanishes in the compression direction and wrinkles mark the region where the compressive stress has collapsed.Two complementary approaches have provided some insight into wrinkled sheets under FFT loading conditions. In a 1961 paper (5), Stein and Hedgepeth computed the asymptotic stress field in infinitely thin sheets under compression by assuming a vanishing component of the stress tensor along the compression direction. They further showed how such an asymptotic stress field yields the extent of wrinkles in several basic examples. A s...

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Section: Far-from-threshold Wrinklingsupporting

confidence: 81%

“…15 by using numerical and asymptotic analysis (15), but to our knowledge the above force balance argument has not been noted before. ‡…”

Section: Near-threshold Wrinklingmentioning

confidence: 86%

“…Here the threshold line was obtained from a linear analysis similar to ref. 15 (albeit with a free edge BC at r ¼ R in to eliminate a spurious boundary layer near r ¼ R in ).…”

Section: Discussionmentioning

confidence: 99%

Section: Near-threshold Wrinklingmentioning

confidence: 99%

See 2 more Smart Citations

Prototypical model for tensional wrinkling in thin sheets

Davidovitch

Schroll

Vella

et al. 2011

Proc. Natl. Acad. Sci. U.S.A.

215

306

View full text Add to dashboard Cite

show abstract

“…However, buckling becomes likely whenever compressive stresses develop in some region of the plate, and this does not require the presence of any compressive external loads. Indeed, this was the situation in some recent works [3,4] where the concerted effect of inhomogeneous pre-buckling deformations and tensile loads led to localized wrinkling patterns (when global tensile loads are involved, it is customary to use the words 'buckling' and 'wrinkling' somewhat interchangeably). Although the resulting linear equations were not solvable in closed form, the presence of initial tension introduced a large nondimensional parameter that played a key role in obtaining analytic formulae for characterizing both the critical load and the buckling pattern.…”

mentioning

confidence: 99%

Asymptotic phenomena in pressurized thin films

2015

Self Cite

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An asymptotic study of the wrinkling of a pressurized circular thin film is performed. The corresponding boundary-value problem is described by two nondimensional parameters; a background tension μ and the applied loading P. Previous numerical studies of the same configuration have shown that P tends to be large, and this fact is exploited here in the derivation of asymptotic descriptions of the elastic bifurcation phenomena. Two limiting cases are considered; in the first, the background tension is modest, while the second deals with the situation when it is large. In both instances, it is shown how the wrinkling is confined to a relatively narrow zone near the rim of the thin film, but the mechanisms driving the bifurcation are different. In the first scenario, the wrinkles are confined to a region which, though close to the rim, is asymptotically separate from it. By contrast, when μ is larger, the wrinkling is within a zone that is attached to the rim. Predictions are made for the value of the applied loading P necessary to generate wrinkling, as well as details of the corresponding wrinkling pattern, and these asymptotic results are compared to some direct numerical simulations of the original boundary-value problem.

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Semi-analytical approximations for a class of multi-parameter eigenvalue problems related to tensile buckling

Coman

Xiang

2012

Z. Angew. Math. Phys.

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On the wrinkling of a pre-stressed annular thin film in tension

Cited by 55 publications

References 17 publications

Prototypical model for tensional wrinkling in thin sheets

Prototypical model for tensional wrinkling in thin sheets

Asymptotic phenomena in pressurized thin films

Semi-analytical approximations for a class of multi-parameter eigenvalue problems related to tensile buckling

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