Crumples as a Generic Stress-Focusing Instability in Confined Sheets

Thin elastic solids are easily deformed into a myriad of three-dimensional shapes, which may contain sharp localized structures as in a crumpled candy wrapper or have smooth and diffuse features like the undulating edge of a flower. Anticipating and controlling these morphologies is crucial to a variety of applications involving textiles, synthetic skins, and inflatable structures. Here, we show that a “wrinkle-to-crumple” transition, previously observed in specific settings, is a ubiquitous response for confined sheets. This unified picture is borne out of a suite of model experiments on polymer films confined to liquid interfaces with spherical, hyperbolic, and cylindrical geometries, which are complemented by experiments on macroscopic membranes inflated with gas. We use measurements across this wide range of geometries, boundary conditions, and length scales to quantify several robust morphological features of the crumpled phase, and we build an empirical phase diagram for crumple formation that disentangles the competing effects of curvature and compression. Our results suggest that crumples are a generic microstructure that emerge at large curvatures due to a competition of elastic and substrate energies.

Popular Summary

Thin sheets permeate our lives, from textiles to biological membranes to synthetic skins. Despite their ubiquity, researchers still face difficulties in predicting how a sheet will respond when it is bent, pulled, or twisted far from its original shape, owing to nonlinearities rooted in geometry. The response can be sharp and localized like the features in a crumpled piece of paper or smooth and diffuse like the undulating edge of a flower. Anticipating the emergence of sharp deformations from smooth confinement remains a major challenge. Here, we provide the experimental basis for a general understanding of this “stress focusing” in thin solids.

We demonstrate a generic route from smooth to sharp deformations in elastic sheets in a suite of experiments ranging from floating polymer films in spherical, hyperbolic, and cylindrical geometries to balloons of various shapes that we inflate with gas. We characterize the morphology of the sharp “crumpled” phase, and we identify the conditions for reaching it. We establish simple scaling relations that unify different geometries and materials. For instance, we show how crumples seen in stiff mylar balloons can be reproduced in rubber balloons in a suitable range of pressures.

Our work shows that smooth wrinkles, which have been widely studied as a platform for metrologies and for engineering smart surfaces, do not survive to large curvatures, where stress-focusing crumples take their place. Our experiments highlight the need for a theoretical understanding of this ubiquitous elastic building block, while providing concrete directions for such studies.

Figure 1

Wrinkle-to-crumple transition in a wide range of geometries, boundary conditions, and system sizes. The overall Gaussian curvature $K$ imposed on the relevant portion of the sheet is given below each schematic. (a) A circular polystyrene sheet ($E=3.4\mathrm{GPa}$) of radius $W=1.5\mathrm{mm}$ and thickness $t=77\mathrm{nm}$ on a liquid meniscus forms crumples at a large droplet curvature [25]. Images adapted from Ref. [26] with permission. (b) A polystyrene sheet floating on a liquid bath forms crumples when indented beyond a threshold depth. Here, $W=11\mathrm{mm}$ and $t=436\mathrm{nm}$. (c) Central portion of a rectangular polystyrene film of thickness $t=157\mathrm{nm}$, width $W=3.3\mathrm{mm}$, and length 9.7 mm on a water meniscus that is uniaxially compressed between two barriers. Wrinkles form when the meniscus is flat or gently curved; crumples form when the imposed curvature along the wrinkle crests is sufficiently large. (d) A square polyethylene bag ($t=102\mu \mathrm{m}$, $E=210\mathrm{MPa}$) forms wrinkles when inflated with sufficient internal pressure $P$. Crumples occur at a lower pressure. The deflated bag width is $W=20\mathrm{cm}$. Supplemental Movies 1–3 in [27] show the transitions in (b)–(d), respectively.

Figure 2

Schematic of a wrinkled rectangular patch, showing ${\sigma }_{\parallel }$ and $R_{\parallel }$. Their values and the total length of the buckled region, ${\ell }_{\parallel }$, are quantified in Table 1 for each setup.

Figure 3

Crumple morphology. (a) Renderings of three-dimensional scans of an inflated square polyethylene bag ($t=49\mu \mathrm{m}$, $E=297\mathrm{MPa}$, $W=20\mathrm{cm}$), showing the topography of wrinkles (top) and crumples (bottom). (b) Cross sections through a single buckled feature as a function of the gauge pressure. The profiles are taken along the dashed lines in (a), and the two circles highlight the sharp tips at the two ends of the crumple. (c) Cross sections along a perpendicular direction, which show only a small change in width through the transition [the same scale as (a),(b)]. (d) Local Gaussian curvature $K$ for the scans shown in (a). The wrinkled state (top) shows large positive and negative curvatures along the wrinkle crests and troughs. The crumpled state (bottom) localizes $K$ and, hence, material stresses. Scale bars in (a)–(d), 1 cm. (e),(f) Origami patterns resembling crumpled states, suggesting that crumples may be an approximate isometry of the plane, with localized bending and stretching in place of the edges and vertices in these paper models.

Figure 4

Emergent scales in the crumpled phase. (a) Crumple length ${\ell }_{\mathrm{cr}}$, measured from optical images in the sheet-on-cylinder, indentation, and inflated membrane setups. The sheet thickness and modulus are varied over a wide range by using rubber ($E\sim 2\mathrm{MPa}$), polyethylene ($E\sim 200\mathrm{MPa}$), and polystyrene ($E=3.4\mathrm{GPa}$) sheets. We also vary the surface tension in the sheet-on-droplet setup (open symbols, $72\mathrm{mN}/\mathrm{m}$; closed symbols, $36\mathrm{mN}/\mathrm{m}$). The data are reasonably described by a simple expression involving ${\ell }_{\parallel }$, $R_{\parallel }$, and ${\sigma }_{\parallel }/Y$ [solid line, Eq. (1)]. The two images show ${\ell }_{\mathrm{cr}}$ increasing upon compression; this behavior may account for some of the scatter in the data. Inset: ${\ell }_{\mathrm{cr}}$ versus $R_{\parallel }$, which does not collapse the data. (b) Minimum angular separation between crumpled regions in indentation. The angular spacing grows with indentation depth $\delta$ and is larger for thicker films.

Figure 5

Phase diagram for wrinkles and crumples. (a) Crumpling threshold measured in the four experimental setups in Fig. 1, including sheet-on-droplet data from King et al. [25] and indentation data from Paulsen et al. [20]. Values of ${\ell }_{\parallel }$, $R_{\parallel }$, and ${\sigma }_{\parallel }$ are either measured or deduced from other measured quantities (see Table 1). The sheet thickness ranges from $40<t<630\mathrm{nm}$ for the polymer films and $15<t<222\mu \mathrm{m}$ for the inflated membranes. Indentation data have $\gamma =72\mathrm{mN}/\mathrm{m}$ and $W$ shown in the legend. Sheet-on-cylinder data have $W=3.2\mathrm{mm}$ and $\gamma$ shown in the legend. Parameters for inflated membranes are detailed in Fig. 6 and 6. The data are reasonably well described by Eq. (2) with $\alpha ={\alpha }_{\mathrm{cr}}=155$ (dashed line). Inset: Threshold curvature $1/R_{\parallel }$ for crumpling, versus wrinkle length ${\ell }_{\parallel }$, in the sheet-on-cylinder setup at fixed $t$ and $\gamma$. Dashed line: Equation (2) with $\alpha =155$. (b) Crumpling threshold versus bendability ${\epsilon }^{-1}$. The threshold is approximately constant over a wide range of bendability. (c) Wrinkling threshold in the indentation setup, measured in the same indentation experiments as (a),(b). The scatter in the filled and open triangles is similar in magnitude to their scatter in (b), respectively.

Figure 6

Pressure threshold for transitioning from sharp crumples to smooth wrinkles in inflated membranes. (a) Experiments using square bags of various thickness ($15<t<222\mu \mathrm{m}$), width ($10<W<31\mathrm{cm}$), and Young’s modulus ($2.0<E<1500\mathrm{MPa}$). Open symbols show the transition for decreasing pressure; the rest of the data are obtained by increasing the pressure. The data are well described by Eq. (3) with ${\alpha }_{\mathrm{cr}}=100$ (dashed line). Inset: ${\ell }_{\parallel }$ and $R_{\parallel }$ versus pressure for a square polyethylene bag ($W=20\mathrm{cm}$, $t=49\mu \mathrm{m}$). Both quantities are relatively constant near the threshold pressure (dashed line, $P_{\mathrm{cr}}$). (b) Experiments with different bag shapes, which are shown in the insets prior to inflation. We use a variety of polyethylene ($E\sim 200\mathrm{MPa}$) and rubber ($E\sim 2\mathrm{MPa}$) sheets to vary the Young’s modulus and thickness, as denoted in the legend. Here, $9.4<W<31\mathrm{cm}$. The crumpling threshold is consistent with the result for square bags, as shown by the gray squares and dashed line that are repeated from (a).

Figure 7

Role of compression in the crumpling threshold. Symbols show the observed crumpling transition for experiments in the sheet-on-cylinder setup, where either the curvature $1/R_{\parallel }$ or the compression $\mathrm{\Delta }$ are varied while the other is held fixed. The sheet width ($1.6<W<3.2\mathrm{mm}$) and thickness ($50<t<300\mathrm{nm}$) are also varied, and $\gamma =72\mathrm{mN}/\mathrm{m}$. Gray bands show the range probed in experiments. Over an intermediate range of $\alpha$, wrinkles or crumples may both exist, depending on the compression. Filled symbols denote a transition to a morphology with crumples but no wrinkles, marked by the darker gray bands. (Open symbols with fixed $\tilde{\mathrm{\Delta }}>0.03$ are also shown in Fig. 5.)