Wrinkle patterns in compressed thin sheets are ubiquitous in nature and technology, from the furrows on our foreheads to crinkly plant leaves, from ripples on plastic-wrapped objects to the protein film on milk. The current understanding of an elementary descriptor of wrinkles-their wavelength-is restricted to deformations that are parallel, spatially uniform, and nearly planar. However, most naturally occurring wrinkles do not satisfy these stipulations. Here we present a scheme that quantitatively explains the wrinkle wavelength beyond such idealized situations. We propose a local law that incorporates both mechanical and geometrical effects on the spatial variation of wrinkle wavelength. Our experiments on thin polymer films provide strong evidence for its validity. Understanding how wavelength depends on the properties of the sheet and the underlying liquid or elastic subphase is crucial for applications where wrinkles are used to sculpt surface topography, to measure properties of the sheet, or to infer forces applied to a film.elastic sheets | wrinkles | curved topography W rinkles emerge in response to confinement, allowing a thin sheet to avoid the high energy cost associated with compressing a fraction e Δ of its length ( Fig. 1) (1-7). The wavelength, λ, of wrinkles reflects a balance between two competing effects: the bending resistance, which favors large wavelengths, and a restoring force that favors small amplitudes of deviation from the flat, unwrinkled state. Two such restoring forces are those due to the stiffness of a solid foundation or the hydrostatic pressure of a liquid subphase (Fig. 1A). Cerda and Mahadevan (1) realized that a tension in the sheet can give rise to a qualitatively similar effect ( Fig. 1B) and thereby proposed a universal law that applies in situations where the wrinkled sheet is nearly planar and subjected to uniaxial loading:Here the bending modulus B = Et 3 =½12ð1 − Λ 2 Þ (with E the Young's modulus, t the sheet's thickness, and Λ the Poisson ratio), whereas out-of-plane deformation is resisted by an effective stiffness, K eff , which can originate from a fluid or elastic substrate, an applied tension, or both. Eq. 1 is appealing in its simplicity, but it applies only for patterns that are effectively one-dimensional. In particular, it does not apply when the stress varies spatially or when there is significant curvature along the wrinkles. Here, we study two experimental settings in which these limitations are crucial: (i) indentation of a thin polymer sheet floating on a liquid, which leads to a horn-shaped surface with negative Gaussian curvature, and (ii) a circular sheet attached to a curved liquid meniscus with positive Gaussian curvature. In both cases, wrinkle patterns live on a curved surface, show spatially varying wavelengths, and are limited in spatial extent. The extent of finite wrinkle patterns in a variety of such 2D situations has recently been addressed (6,(8)(9)(10)(11) and was found to depend largely on external forces and boundary conditions. Howeve...

Numerically simulating deformations in thin elastic sheets is a challenging problem in computational mechanics due to destabilizing compressive stresses that result in wrinkling. Determining the location, structure, and evolution of wrinkles in these problems have important implications in design and is an area of increasing interest in the fields of physics and engineering. In this work, several numerical approaches previously proposed to model equilibrium deformations in thin elastic sheets are compared. These include standard finite element-based static post-buckling approaches as well as a recently proposed method based on dynamic relaxation, which are applied to the problem of an annular sheet with opposed tractions where wrinkling is a key feature. Numerical solutions are compared to analytic predictions, enabling a quantitative evaluation of the predictive power of the various methods. Results indicate that static finite element approaches are highly sensitive to initial imperfections, relying on a priori knowledge of the equilibrium wrinkling pattern to generate optimal results. In contrast, dynamic relaxation is much less sensitive to initial imperfections and can generate solutions for a wide variety of loading conditions without requiring knowledge of the equilibrium solution beforehand.

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...

Exogenous ferricyanide is reduced by roots of Z. mays. In contrast to oxidation of exogenous electron donors, ferricyanide reduction occurs mostly at the apical 5 mm of the root. Using just this portion of the root, it is shown that the activity is neither a consequence of uptake of ferricyanide followed by excretion of its reduced form, nor of leakage of a reductant. Addition of ferricyanide for 40 s or 5 min results in an apparent oxidation of NADPH but not of NADH; rates of ferricyanide reduction vary together with levels of NADPH but not of NADH in the presence or absence of oxygen. It is concluded that an enzyme which can oxidize cytoplasmic NADPH and transfer the electrons to an external acceptor exists at the cell surface of maize roots. This finding extends the results of others who showed similar redox activity at the surface of Fe-depleted dicotyledonous roots, and indicates that an energy source other than ATP exists at the cell surface of a variety of plants under unstressed conditions.

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