Wrinkles, rucks and folds formed in a heavy sheet on a frictional surface

TL;DR

This work investigates how gravity, elasticity, and interfacial friction shape the indentation-induced morphologies of heavy elastic sheets on rigid substrates. Using a center-indentation protocol, the authors combine experiments, finite-element simulations, and Föppl–von Kármán theory to map a sequence from axisymmetric uplift through wrinkles to global buckling and unloading folds, revealing universal and friction-modulated behaviors. In the frictionless case, wrinkle number is effectively fixed (about $m\approx7$) and the onset displacement scales with sheet thickness via the elasto-gravitational length $\ell_g=(B/(\rho g))^{1/4}$, while friction introduces a nondimensional parameter $\tau=\mu a h/\ell_g^2$ that shifts wrinkle count and onset in a near-threshold regime. The study also derives a global-buckling criterion $d_c/h\sim a/\ell_g$ and identifies Type-I versus Type-II wrinkle regimes, with hysteresis and irreversible folding upon unloading in certain loading paths. Together, these results provide a simple framework for programmable sheet morphogenesis driven by gravity and friction, with potential applications in design of wrinkles and folds in thin films and architectural membranes.

Abstract

Soft elastic sheets resting on rigid surfaces develop wrinkles, rucks, and folds due to the combined influence of elasticity, gravity, and contact interactions. Despite their ubiquity, the principles governing their morphology and transitions remain unclear. We introduce a minimal experiment in which the center of a gravity-loaded sheet is gradually lifted from the supporting plane. This operation generates a clear sequence of shapes: an axisymmetric uplift, a finite number of wrinkles, system-spanning rucks produced by global buckling, and folded states that can arise from ruck collapse upon unloading at larger lifts. Combining experiments, finite-element simulations, and Föppl-von Kármán theory, we establish a unified physical picture of this morphology sequence. In the frictionless case, elasticity and gravity alone govern the response, leading to a universal wrinkling threshold: the wrinkle number is fixed and the onset displacement scales linearly with the sheet thickness. With interfacial friction, the wrinkled state is described by introducing an additional nondimensional parameter that compares frictional and elastic-gravitational forces. These results suggest a simple route to programmable sheet morphogenesis via friction and gravity.

Figures (13)

• Figure 1: (a) Halloween ghost formed by draping a soft, thin fabric over a convex object. (b) Schematic of the experimental setup. A thin elastic sheet of radius $a$ is indented vertically at its center by a distance $d$ from beneath the sheet on a rigid flat substrate. A region with a certain radius $R$ detaches from the substrate. The indentation force $F$ is measured as a function of the indentation displacement $d$ by a load cell attached to the indenter. (c) Representative images of three distinct lifted shapes for increasing $d$: (i) axisymmetric, (ii) wrinkled, and (iii) globally buckled. (d, e): Color maps of the vertical displacements $z=w(x,y)$ for the indicated values of $d$. (d) The point-cloud data acquired by a 3D scanner in our experiments and (e) those in our FES. The parameters used both in experiment and FES are $E=477\, {\rm kPa}, \nu=0.47, \rho=1127\, {\rm kg/m}^3, h = 0.27\, {\rm mm}, a = 104\, {\rm mm}$ and $\mu=0.32.$
• Figure 2: Lifting force and lifted radius vs. indentation height. The left axis and red points indicate the lifting force $F$, normalized by the total weight of the sheet $Mg$. The lifted radius $R$, normalized by the sheet’s full radius $a$, is plotted on the right axis (blue points). Filled and open symbols represent data obtained from experiments and FES, respectively. The data are taken from the experiment and simulation presented in Fig. \ref{['Figure01']}. Inset figures show lifted shapes of the sheet obtained from our FES. Axisymmetry breaks at $d=d_w$, where $m$-fold wrinkles emerge ($m$=8 and $d_w\approx 4.9$ mm). The critical displacement for global buckling is $d_c\approx 9.2$ mm.
• Figure 3: Dimensionless (a) lifted radius $R/\ell_g$ and (b) lifting force $F/(\rho g h \ell_g^2)$ plotted against normalized central displacement $d/h$. The characteristic length $\ell_g$ is defined in Eq. (\ref{['sec03:def:ellg']}). Filled and open symbols denotes experimental and FES data, respectively. The scaling behaviors predicted by Eqs. (\ref{['sec03:eq:R']}) and (\ref{['sec03:eq:F']}) are confirmed in both the shallow ($d\ll h$) and large indentation ($d\gg h$) regimes, as shown by the dotted and solid lines with the coefficient in Eqs. (\ref{['sec04A:sol:kb']}, \ref{['sec03:eq:Fd_bend_w_pre']}, \ref{['sec04B:sol:cs']}, and \ref{['sec04B:sol:ks']}).
• Figure 4: The profile of the azimuthally averaged stress $h\overline{\sigma_{\alpha\beta}}(r)=\frac{1}{2\pi}\oint h\sigma_{\alpha\beta}(r, \theta) d\theta$ is plotted normalized by the characteristic stress scale $B/\ell_g^2$. Dots and solid lines represent a FES result based on the simulation described in Fig. \ref{['Figure01']} (with $d=4.66~\textrm{mm}$) and our theory in § \ref{['Sec:theory_liftedregion']}, respectively. The shaded region corresponds to the contact region.
• Figure 5: Displacement field in the contact region for (i) small indentation height ($d = 0.86$ mm) and (ii) large indentation height ($d = 4.66$ mm). (a) Quarter of the sheet is shown as the shaded area. The color of the inward-pointing vectors represents the magnitude of the displacement $\bm{u}(r, \theta)$ in the contact region, normalized by the representative mesh size $\Delta x$ (here equals to $1~\textrm{mm}$) of the finite elements. (b) Radial profiles of the azimuthally averaged radial displacement, defined as $\overline{u_r}(r) \equiv \frac{1}{2\pi} \oint u_r(r, \theta) d\theta$, plotted as a function of $r/R$. The solid and dashed reference lines indicate $\overline{u_r}/\Delta x = 0$ and $-10^{-3}$, respectively. (c) Radial displacement at the edge of the sheet, $\overline{u_r}(a)$, as a function of $d$. The values of $\overline{u_r}(a)$ are extracted from the red circle markers in panel (b). A sufficiently large negative value of $\overline{u_r}(a)$ indicates that the entire sheet has slipped. In the parameter ranges explored in this study, almost of the sheet slip completely before the onset of wrinkling. These results are obtained from the finite-element simulations shown in Fig. \ref{['Figure01']}.