Wrinkling and developable cones in centrally confined sheets.

Thin sheets respond to confinement by smoothly wrinkling or by focusing stress into small, sharp regions. From engineering to biology, geology, textiles, and art, thin sheets are packed and confined in a wide variety of ways, and yet fundamental questions remain about how stresses focus and patterns form in these structures. Using experiments and molecular dynamics simulations, we probe the confinement response of circular sheets, flattened in their central region and quasistatically drawn through a ring.

Self-Ordering of Buckling, Bending, and Bumping Beams.

A collection of thin structures buckle, bend, and bump into each other when confined. This contact can lead to the formation of patterns: hair will self-organize in curls; DNA strands will layer into cell nuclei; paper, when crumpled, will fold in on itself, forming a maze of interleaved sheets. This pattern formation changes how densely the structures can pack, as well as the mechanical properties of the system.

Elastogranular columns and beams.

String and grains can be combined to create structures capable of bearing significant loads. In this work, we prepare columns and beams through a layer-by-layer deposition of granular matter and loops of fiber strings, and characterize their mechanical properties. The loops cause the grains to jam, and the inter-grain contact leads to a Hertzian-like constitutive response.

Efficient snap-through of spherical caps by applying a localized curvature stimulus.

In bistable actuators and other engineered devices, a homogeneous stimulus (e.g., mechanical, chemical, thermal, or magnetic) is often applied to an entire shell to initiate a snap-through instability.

Elastic Instabilities Govern the Morphogenesis of the Optic Cup.

Because the normal operation of the eye depends on sensitive morphogenetic processes for its eventual shape, developmental flaws can lead to wide-ranging ocular defects. However, the physical processes and mechanisms governing ocular morphogenesis are not well understood. Here, using analytical theory and nonlinear shell finite-element simulations, we show, for optic vesicles experiencing matrix-constrained growth, that elastic instabilities govern the optic cup morphogenesis.

Emergence of structure in columns of grains and elastic loops.

It is possible to build free-standing, load-bearing structures using only rocks and loops of elastic material. We investigate how these structures emerge, and find that the necessary maximum loop spacing (the critical spacing) is a function of the frictional properties of the grains and the elasticity of the confining material. We derive a model to understand both of these relationships, which depends on a simplification of the behavior of the grains at the edge of a structure.

Grasping with kirigami shells.

The ability to grab, hold, and manipulate objects is a vital and fundamental operation in biological and engineering systems. Here, we present a soft gripper using a simple material system that enables precise and rapid grasping, and can be miniaturized, modularized, and remotely actuated. This soft gripper is based on kirigami shells-thin, elastic shells patterned with an array of cuts.

Nonlinear buckling behavior of a complete spherical shell under uniform external pressure and homogenous natural curvature.

In this work, we consider the stability of a spherical shell under combined loading from a uniform external pressure and a homogenous natural curvature. Nonmechanical stimuli, such as one that tends to modify the rest curvature of an elastic body, are prevalent in a wide range of natural and engineered systems, and may occur due to thermal expansion, changes in pH, differential swelling, and differential growth. Here we investigate how the presence of both an evolving natural curvature and an external pressure modifies the stability of a complete spherical shell.

Packing transitions in the elastogranular confinement of a slender loop.

Confined thin structures are ubiquitous in nature. Spatial and length constraints have led to a number of novel packing strategies at both the micro-scale, as when DNA packages inside a capsid, and the macro-scale, seen in plant root development and the arrangement of the human intestinal tract. Here, we investigate the resulting packing behaviors between a growing slender structure constrained by deformable boundaries.

Evolution of critical buckling conditions in imperfect bilayer shells through residual swelling.

We propose and investigate a minimal mechanism that makes use of differential swelling to modify the critical buckling conditions of elastic bilayer shells, as measured by the knockdown factor. Our shells contain an engineered defect at the north pole and are made of two layers of different crosslinked polymers that exchange free molecular chains. Depending on the size of the defect and the extent of swelling, we can observe either a decreasing or increasing knockdown factor.

Buckling of geometrically confined shells.

We study the periodic buckling patterns that emerge when elastic shells are subjected to geometric confinement. Residual swelling provides access to range of shapes (saddles, rolled sheets, cylinders, and spherical sections) which vary in their extrinsic and intrinsic curvatures. Our experimental and numerical data show that when these moderately thick structures are radially confined, a single geometric parameter - the ratio of the total shell radius to the amount of unconstrained material - predicts the number of lobes formed.

Static bistability of spherical caps.

Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two 'self-equilibrated' states, one natural and the other inside out (or 'everted'). Though this is familiar from everyday life-an umbrella is remarkably stable, yet a contact lens can be easily turned inside out-the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite-element simulations to determine the threshold between bistability and monostability for shells of different solid angle.

Elastogranular Mechanics: Buckling, Jamming, and Structure Formation.

Confinement of a slender body into a granular array induces stress localization in the geometrically nonlinear structure, and jamming, reordering, and vertical dislodging of the surrounding granular medium. By varying the initial packing density of grains and the length of a confined elastica, we identify the critical length necessary to induce jamming, and demonstrate how folds couple with the granular medium to localize along grain boundaries. Above the jamming threshold, the characteristic length of elastica deformation is shown to diverge in a manner that is coupled with the motion and rearrangement of the grains, suggesting the ordering of the granular array governs the deformation of the slender structure.

Curvature-Induced Instabilities of Shells.

Induced by proteins within the cell membrane or by differential growth, heating, or swelling, spontaneous curvatures can drastically affect the morphology of thin bodies and induce mechanical instabilities. Yet, the interaction of spontaneous curvature and geometric frustration in curved shells remains poorly understood. Via a combination of precision experiments on elastomeric spherical shells, simulations, and theory, we show how a spontaneous curvature induces a rotational symmetry-breaking buckling as well as a snapping instability reminiscent of the Venus fly trap closure mechanism.

Extended lubrication theory: improved estimates of flow in channels with variable geometry.

Lubrication theory is broadly applicable to the flow characterization of thin fluid films and the motion of particles near surfaces. We offer an extension to lubrication theory by starting with Stokes equations and considering higher-order terms in a systematic perturbation expansion to describe the fluid flow in a channel with features of a modest aspect ratio. Experimental results qualitatively confirm the higher-order analytical solutions, while numerical results are in very good agreement with the higher-order analytical results.

Kirigami actuators.

Thin elastic sheets bend easily and, if they are patterned with cuts, can deform in sophisticated ways. Here we show that carefully tuning the location and arrangement of cuts within thin sheets enables the design of mechanical actuators that scale down to atomically-thin 2D materials. We first show that by understanding the mechanics of a single non-propagating crack in a sheet, we can generate four fundamental forms of linear actuation: roll, pitch, yaw, and lift.

Revisiting the generalized scaling law for adhesion: role of compliance and extension to progressive failure.

A generalized scaling law, based on the classical fracture mechanics approach, is developed to predict the bond strength of adhesive systems. The proposed scaling relationship depends on the rate of change of debond area with compliance, rather than the ratio of area to compliance. This distinction can have a profound impact on the expected bond strength of systems, particularly when the failure mechanism changes or the compliance of the load train increases.

Geometry and mechanics of thin growing bilayers.

We investigate how thin sheets of arbitrary shapes morph under the isotropic in-plane expansion of their top surface, which may represent several stimuli such as nonuniform heating, local swelling and differential growth. Inspired by geometry, an analytical model is presented that rationalizes how the shape of the disk influences morphing, from the initial spherical bending to the final isometric limit. We introduce a new measure of slenderness that describes a sheet in terms of both thickness and plate shape.

Rising beyond elastocapillarity.

We consider the elastocapillary rise between swellable structures using a favorable solvent. We characterize the dynamic deformations and resulting equilibrium configurations for various beams. Our analysis reveals the importance of the spacing between the two beams, and the elastocapillary length lec, which prescribes the relative magnitude of surface tension and bending stiffness in the system.

Morphing of geometric composites via residual swelling.

Understanding and controlling the shape of thin, soft objects has been the focus of significant research efforts among physicists, biologists, and engineers in the last decade. These studies aim to utilize advanced materials in novel, adaptive ways such as fabricating smart actuators or mimicking living tissues. Here, we present the controlled growth-like morphing of 2D sheets into 3D shapes by preparing geometric composite structures that deform by residual swelling.

Buckling of dielectric elastomeric plates for soft, electrically active microfluidic pumps.

Elastic instabilities, when properly implemented within soft, mechanical structures, can generate advanced functionality. In this work, we use the voltage-induced buckling of thin, flexible plates to pump fluids within a microfluidic channel. The soft electrodes that enable electrical actuation are compatible with fluids, and undergo large, reversible deformations.

Swelling-induced deformations: a materials-defined transition from macroscale to microscale deformations.

Swelling-induced deformations are common in many biological and industrial environments, and the shapes and patterns that emerge can vary across many length scales. Here we present an experimental study of a transition between macroscopic structural bending and microscopic surface creasing in elastomeric beams swollen non-homogeneously with favorable . We show that this transition is dictated by the materials and geometry of the system, and we develop a simple scaling model based on competition between bending and swelling energies that predicts if a given droplet would deform a polymeric structure macroscopically or microscopically.

Mechanics of surface area regulation in cells examined with confined lipid membranes.

Cells are wrapped in inelastic membranes, yet they can sustain large mechanical strains by regulating their area. The area regulation in cells is achieved either by membrane folding or by membrane exo- and endocytosis. These processes involve complex morphological transformations of the cell membrane, i.

Draping films: a wrinkle to fold transition.

A polymer film draping over a point of contact will wrinkle due to the strain imposed by the underlying substrate. The wrinkle wavelength is dictated by a balance of material properties and geometry; most directly the thickness of the draping film. At a critical strain, the stress in the film will localize, causing hundreds of wrinkles to collapse into several discrete folds.

Crumpled surface structures.

We present a scalable patterning method based on surface plate buckling, or crumpling, to generate a variety of topographies that can dynamically change shape and aspect ratio in response to stimuli.