Geometry-Driven Folding of a Floating Annular Sheet

Faceted wrinkling by contracting a curved boundary

Single-mode deformations of two-dimensional materials, such as the Miura-ori zig-zag fold, are important to the design of deployable structures because of their robustness; these usually require careful pre-patterning of the material. Here we show that inward contraction of a curved boundary produces a fine wrinkle pattern with a novel structure that suggests similar single-mode characteristics, but with minimal pre-patterning. Using finite-element representation of the contraction of a thin circular annular sheet, we show that these sheets wrinkle into a structure well approximated by an isometric structure composed of conical sectors and flat, triangular facets. Isometry favours the restriction of such deformations to a robust low-bending energy channel that avoids stretching. This class of buckling offers a novel way to manipulate sheet morphology via boundary forces.

Crystal capillary origami capsule with self-assembled nanostructures

The self-assembling mechanism of elasto-capillaries opens new applications in micro and nanotechnology by providing 3D assembly structures with 2D planar unit cells, so-called capillary origami. To date, the final structure has been designed based on the predetermined shape and size of the unit cell. Here, we show that plate-like salt crystallites grow and cover the emulsion interface, which is driven by Laplace pressure. Eventually, it creates a spherical capsule with self-assembled nanostructures. The capsule and the crystallite are investigated by scanning electron microscopy and X-ray diffraction analysis. To explain the mechanism, we develop a theoretical model to estimate the capsule size, the shell thickness, and the conditions necessary to form the shell based on a thin-walled pressure vessel. The proposed crystal capillary origami can fabricate a three-dimensional self-assembled salt capsule without any complicated procedures. We believe that it can offer a new physicochemical avenue for the spontaneous and facile fabrication of water-soluble carrier particles.

Dynamic Buckling of an Elastic Ring in a Soap Film

Dynamic buckling may occur when a load is rapidly applied to, or removed from, an elastic object at rest. In contrast to its static counterpart, dynamic buckling offers a wide range of accessible patterns depending on the parameters of the system and the dynamics of the load. To study these effects, we consider experimentally the dynamics of an elastic ring in a soap film when part of the film is suddenly removed. The resulting change in tension applied to the ring creates a range of interesting patterns that cannot be easily accessed in static experiments. Depending on the aspect ratio of the ring's cross section, high-mode buckling patterns are found in the plane of the remaining soap film or out of the plane. Paradoxically, while inertia is required to observe these nontrivial modes, the selected pattern does not depend on inertia itself. The evolution of this pattern beyond the initial instability is studied experimentally and explained through theoretical arguments linking dynamics to pattern selection and mode growth. We also explore the influence of dynamic loading and show numerically that, by imposing a rate of loading that competes with the growth rate of instability, the observed pattern can be selected and controlled.

Geometry underlies the mechanical stiffening and softening of an indented floating film

A basic paradigm underlying the Hookean mechanics of amorphous, isotropic solids is that small deformations are proportional to the magnitude of external forces. However, slender bodies may undergo large deformations even under minute forces, leading to nonlinear responses rooted in purely geometric effects. Here we study the indentation of a polymer film on a liquid bath. Our experiments and simulations support a recently-predicted stiffening response [D. Vella and B. Davidovitch, Phys. Rev. E, 2018, 98, 013003], and we show that the system softens at large slopes, in agreement with our theory that addresses small and large deflections. We show how stiffening and softening emanate from nontrivial yet generic features of the stress and displacement fields.

Programming curvilinear paths of flat inflatables

Inflatable structures offer a path for light deployable structures in medicine, architecture, and aerospace. In this study, we address the challenge of programming the shape of thin sheets of high-stretching modulus cut and sealed along their edges. Internal pressure induces the inflation of the structure into a deployed shape that maximizes its volume. We focus on the shape and nonlinear mechanics of inflated rings and more generally, of any sealed curvilinear path. We rationalize the stress state of the sheet and infer the counterintuitive increase of curvature observed on inflation. In addition to the change of curvature, wrinkles patterns are observed in the region under compression in agreement with our minimal model. We finally develop a simple numerical tool to solve the inverse problem of programming any 2-dimensional (2D) curve on inflation and illustrate the application potential by moving an object along an intricate target path with a simple pressure input.

Hierarchical wrinkles and oscillatory cracks in metal films deposited on liquid stripes

Fascinating crack and wrinkle patterns driven by stresses are ubiquitous in natural and artificial systems. It is of great interest to control the morphologies of stress-driven patterns by using facile techniques. Here we report on the spontaneous formation of hierarchical wrinkles and oscillatory cracks in metal films deposited on liquid (or soft polymer) stripes. It is found that the metal film is under a tensile stress during deposition owing to the thermal expansion of the liquid substrate. As the film thickness is beyond a critical value, oscillatory cracks with sawtoothlike shapes form on the liquid stripes. The ratio of crack oscillation period to amplitude is independent of the stripe width and film material, which can be well explained by the "brittle adhesive joints" model. After deposition, the metal film is under a compressive stress, which is relieved by formation of various wrinkle patterns. Hierarchical wrinkles with changing wavelengths form near the stripe edge while labyrinth or wavy wrinkles form at the center. Energy analysis was adopted to explain the formation and evolution of the wrinkle patterns. This study could promote better understanding of the formations of crack and wrinkle patterns in constrained film structures and controllable fabrication of stress-driven patterns by prefabricating liquid (or soft polymer) interlayer arrays.

Fluid-supported elastic sheet under compression: Multifold solutions

The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.

Geometrically incompatible confinement of solids

The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current-day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's elastica-a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and that has been widely applied to classical and modern studies of beams and rods. We define a class of geometrically incompatible confinement problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' Theorema Egregium, induces strain. By focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler elastica" This emergent rule states that-despite the unavoidable strain in such an incompatible confinement-the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler elastica underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.

Microscopic details of a fluid/thin film triple line

In recent years, there has been a considerable interest in the mechanics of soft objects meeting fluid interfaces (elasto-capillary interactions). In this work we experimentally examine the case of a fluid resting on a thin film of rigid material which, in turn, is resting on a fluid substrate. To simplify complexity, we adapt the experiment to a one-dimensional contact geometry and examine the behaviour of polystyrene and polycarbonate films directly with confocal microscopy. We find that the fluid meets the film in a manner consistent with the Young-Dupré equation when the film is thick, but transitions to what appears similar to a Neumann-like balance when the thickness is decreased. However, on closer investigation we find that the true contact angle is always given by the Young construction. The apparent paradox is a result of macroscopically measured angles not being directly related to true microscopic contact angles when curvature is present. We model the effect with an Euler-Bernoulli beam on a Winkler foundation as well as with an equivalent energy-based capillary model. Notably, the models highlight several important lengthscales and the complex interplay of tension, gravity, and bending in the problem.

Cylinder morphology of a stretched and twisted ribbon

A rich zoology of morphologies emerges from a simple stretched and twisted elastic ribbon. Despite a lot of interest, all the observed shapes are not quantitatively described. This is the case of the cylindrical shape that prevails at large tension and twist, which emerges from a transverse buckling instability of the helicoid. Here, we propose a simple description of this cylindrical shape. By comparing its energy to the energy of other configurations, helicoidal and facetted, we are able to determine its location on the tension-twist phase diagram. The theoretical predictions are in good quantitative agreement with the experimental results and complement previous results from linear stability analysis.

Harnessing fold-to-wrinkle transition and hierarchical wrinkling on soft material surfaces by regulating substrate stiffness and sputtering flux

Strain-induced complex surface patterns such as wrinkles, folds and hierarchical structures are quite useful in a wide range of practical applications. Although various surface patterns have been extensively investigated, precisely controlling the coexistence and transition of multimodal structures is still a challenge. In this work, we report on a facile technique to harness fold-to-wrinkle transition and hierarchical wrinkling on soft material surfaces by regulating substrate stiffness and sputtering flux. It is found that as the substrate stiffness or sputtering flux increases, the surface patterns successively evolve from networked folds to isolated folds (coexistence of folds and wrinkles) and finally to labyrinth-like wrinkles. For larger sputtering flux, two distinct wrinkling patterns (i.e., G1 wrinkling due to surface instability during sputtering and G2 wrinkling due to thermal compression after deposition) can coexist on the sample surfaces, resulting in the spontaneous formation of hierarchical wrinkles. The report in this work could promote better understanding of the sputtering effect on the spontaneous pattern formation for soft materials and controllable fabrication of multiple complex patterns by simply regulating substrate stiffness and sputtering flux.

Regimes of wrinkling in an indented floating elastic sheet

A thin, elastic sheet floating on the surface of a liquid bath wrinkles when poked at its center. We study the onset of wrinkling as well as the evolution of the pattern as indentation progresses far beyond the wrinkling threshold. We use tension field theory to describe the macroscopic properties of the deformed film and show that the system passes through a host of different regimes, even while the deflections and strains remain small. We show that the effect of the finite size of the sheet ultimately plays a key role in determining the location of the wrinkle pattern, and obtain scaling relations that characterize the number of wrinkles at threshold and its variation as the indentation progresses.

Partial wetting of thin solid sheets under tension

We consider the equilibrium of liquid droplets sitting on thin elastic sheets that are subject to a boundary tension and/or are clamped at their edge. We use scaling arguments, together with a detailed analysis based on the Föppl-von-Kármán equations, to show that the presence of the droplet may significantly alter the stress locally if the tension in the dry sheet is weak compared to an intrinsic elasto-capillary tension scale γ2/3(Et)1/3 (with γ the droplet surface tension, t the sheet thickness and E its Young modulus). Our detailed analysis suggests that some recent experiments may lie in just such a "non-perturbative" regime. As a result, measurements of the tension in the sheet at the contact line (inferred from the contact angles of the sheet with the liquid-vapour interface) do not necessarily reflect the true tension within the sheet prior to wetting. We discuss various characteristics of this non-perturbative regime.