Geometry-Driven Folding of a Floating Annular Sheet

Abstract

Predicting the large-amplitude deformations of thin elastic sheets is difficult due to the complications of self-contact, geometric nonlinearities, and a multitude of low-lying energy states. We study a simple two-dimensional setting where an annular polymer sheet floating on an air-water interface is subjected to different tensions on the inner and outer rims. The sheet folds and wrinkles into many distinct morphologies that break axisymmetry. These states can be understood within a recent geometric approach for determining the gross shape of extremely bendable yet inextensible sheets by extremizing an appropriate area functional. Our analysis explains the remarkable feature that the observed buckling transitions between wrinkled and folded shapes are insensitive to the bending rigidity of the sheet.The mechanics of thin sheets at fluid interfaces is a current frontier of elasto-capillary phenomena [1]. In contrast to thicker films that balance the liquid-vapor surface tension γ by generating moderate strain [2][3][4][5] or curvature [6][7][8][9], very thin sheets strongly resist in-plane stretching but are readily curled, wrinkled, or folded under capillary forces [10,11]. In the asymptotic regimewhere B, Y are the bending and stretching moduli and R is a characteristic length, the liquid surface energy becomes the only dominant energy, rendering the elastocapillary problem into a purely geometric area minimization. This nontrivial class of "asymptotic isometries" was demonstrated in the partial wrapping of a liquid drop by an ultrathin circular sheet, where axial symmetry is spontaneously broken [12]. Our understanding of this field is still in its infancy, and many basic questions remain. What classes of gross shapes are possible, and what is the nature of the transitions between them? In general, transitions in microstructure -such as the wrinkle-fold transition in 1D systems [13][14][15] -are driven by competing energies. Are there situations in which microstructure is dictated by geometrical, rather than mechanical constraints?Here we study a simple, near-planar system which exhibits: (i) a variety of gross shapes with continuous and discontinuous transitions between them, (ii) coexistence of distinct microstructural elements, and (iii) a wrinklefold transition governed by geometric constraints. We find that purely geometric considerations determine the gross shape, which may dictate a specific microstructure. If more than one microstructure is possible, then mechanical energies may select one.Experiment.-We work in a geometry first experimentally investigated by Piñeirua et al. [16], but with much thinner films (t ∼ 100 nm) in order to probe the asymp- totic regime of Eq. (1). We spin-coat polystyrene films (E = 3.4 GPa) on glass substrates, and cut into an annular shape with radii R in and R out (Fig. 1a), where 1.2 mm < R in < 5.7 mm, and 6.5 mm < R out < 10.5 mm.The film is floated onto water in a Langmuir trough, and surfactant (perfluorododecanoic acid) is added outside the film. Surfactant concentr...

Wrapping Liquids, Solids, and Gases in Thin Sheets

“…When a floating annularshaped film is subjected to a sufficiently large tension on its inner boundary compared to the tension on its outer boundary, static forces fall out of balance and the sheet must contract radially inwards. Paulsen et al (94) showed that for a sufficiently thin film, folding occurs at a threshold ratio of inner to outer tension that depends only on a single geometric parameter: the ratio of the inner to the outer radius. Furthermore, they showed that by changing the size of the hole in the sheet, the transition can be continuous or discontinuous.…”

An asymptotic variational problem modeling a thin elastic sheet on a liquid, lifted at one end

“…x)ds (7) among x, such that (x(0), (0)) is fixed and lim s→∞ (x(s), (s)) = (∞, 0) has a solution that is unique up to reparametrization.…”

“…Using the previous results, we have that d dx is decreasing and nonpositive. Since (x(s), (s)) is a critical point of (2), we have that (x) is a critical point (and therefore minimizer) of (7), and therefore (x) solves…”

“…In order to prove this, we will need a technical remark about the structure of a minimizing sequence of the problem given by (7).…”

Cylinder morphology of a stretched and twisted ribbon

“…Thin elastic sheets exhibit a wide variety of patterns in response to external loadings such as wrinkles, crumples, and folds [1][2][3][4][5][6]. This rich behavior stems from their two dimensional nature, which introduces a coupling between mechanics and geometry.…”