Prototypical model for tensional wrinkling in thin sheets

Abstract

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...

Capillary Deformations of Bendable Films

“…In the high bendability regime the sheet supports very small level of compression before and after buckling. As a consequence, the elastic stresses in the sheet σ rr (r), σ θθ (r) are determined byγ (and γ/γ , ϑ Y ), and exhibit weak, sub-dominant dependence on the bending modulus (hence on ) [15]. This feature distinguishes our system from [6], where bending forces are dominant and balance surface tension.…”

“…2 for ϑ Y = π/2, a few distinct values of γ/γ , and a range ofγ 1. The confinement parameters, α and τ express the degree of variation of the radial tension across each part of the film, which gives rise to compressive azimuthal (hoop) strain [15,18]. The size of the wrinkled zone thus increases with α and τ .…”

“…The parameterK −1 quantifies the deformability of the liquid foundation, andR is the ratio between the sizes of the drop and the sheet. In this study we assumeK,R → 0, corresponding to a sufficiently small drop; the effect ofK,R = 0 on the stress is perturbative [15,16]. The final pair of parameters, essential for our study, involves the elastic modulii of the sheet:…”

“…2). The parameter −1 is the bendability [15], and can be expressed as the ratio between the lengths R and B/γ (often called the "elasto-capillary length" [6]). In the high bendability regime the sheet supports very small level of compression before and after buckling.…”

Disappearance of stretch-induced wrinkles of thin sheets: A study of orthotropic films

“…Although this contribution is an interesting experimental challenge, in this paper we focus on emerging orthotropy and the experimental validation of the disappearance of wrinkles. There are a few publication of related experimental results, some of them uses highly elaborated experimental settings (e.g [6,8]), none of them point out the existence of the second bifurcation point a the resumed flat state of the film. Regarding that lack of observation it is worthy to point out, that even a simple experimental arrangement reproduces the anticipated behavior in a reliable manner due to the well-chosen material and the prestressing.…”