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. Wrinkles develop in the outer, free region, then are replaced by a truncated cone, which forms in an abrupt transition to stress focusing. We explore how the force associated with this event, and the number of wrinkles, depend on geometry. Additional cones sequentially pattern the sheet until axisymmetry is recovered in most geometries. The cone size is sensitive to in-plane geometry. We uncover a coarse-grained description of this geometric dependence, which diverges depending on the proximity to the asymptotic d-cone limit, where the clamp size approaches zero. This paper contributes to the characterization of general confinement of thin sheets, while broadening the understanding of the d cone, a fundamental element of stress focusing, as it appears in realistic settings.

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. How and when these patterns form, as well as the force required to pack these structures is not currently understood. Here we study the emergence of order in a canonical example of packing in slender structures, i.e., a system of parallel confined elastic beams. Using tabletop experiments, simulations, and standard theory from statistical mechanics, we predict the amount of confinement (growth or compression) of the beams that will guarantee a global system order, which depends only on the initial geometry of the system. Furthermore, we find that the compressive stiffness and stored bending energy of this metamaterial are directly proportional to the number of beams that are geometrically frustrated at any given point. We expect these results to elucidate the mechanisms leading to pattern formation in these kinds of systems and to provide a new mechanical metamaterial, with a tunable resistance to compressive force.

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. Initially, one force chain that propagates vertically through the column bears most of the compressive load. As the magnitude of the load is increased, more force chains form in the column, which act in parallel to increase its stiffness, akin to a "super-Hertzian" regime. Applying a compressive prestress enables the structures to withstand shear, enabling the fabrication of cantilevered beams. This work provides a mechanical framework to use elastogranular jamming to create rapid, reusable infrastructure components, such as columns, beams, and arches from inexpensive, commonplace materials, such as rocks and string.

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. In this work, we demonstrate that restricting the active area to the shell boundary allows for a large reduction in its size, thereby decreasing the energy input required to actuate the shell. To do so, we combine theory with 1D finite element simulations of spherical caps with a non-homogeneous distribution of stimulus-responsive material. We rely on the effective curvature stimulus, i.e., the natural curvature induced by the non-mechanical stimulus, which ensures that our results are entirely stimulus-agnostic. To validate our numerics and demonstrate this generality, we also perform two sets of experiments, wherein we use residual swelling of bilayer silicone elastomers-a process that mimics differential growth-as well as a magneto-elastomer to induce curvatures that cause snap-through. Our results elucidate the underlying mechanics, offering an intuitive route to optimal design for efficient snap-through.

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. By capturing the stress amplification owing to mass increase during growth, we show that the morphogenesis is driven by two elastic instabilities analogous to the snap through in spherical shells, where the second instability is sensitive to the optic cup geometry. In particular, if the optic vesicle is too slender, it will buckle and break axisymmetry, thus, preventing normal development. Our results shed light on the morphogenetic mechanisms governing the formation of a functional biological system and the role of elastic instabilities in the shape selection of soft biological structures.

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. We find that higher friction leads to larger stable grain-grain and grain-loop contact angles resulting in a simple function for the frictional critical spacing, which depends linearly on friction to first order. On the other hand, a higher bending rigidity enables the loops to better contain the hydrostatic pressure of the grains, which we understand using a hydroelastic scale. These findings will illuminate the stabilization of dirt by plant roots, and potentially enable the construction of simple adhesion-less structures using only granular material and fiber.

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. The kirigami cut pattern is determined by evaluating the shell's mechanics and geometry, using a combination of experiments, finite element simulations, and theoretical modeling, which enables the gripper design to be both scalable and material independent. We demonstrate that the kirigami shell gripper can be readily integrated with an existing robotic platform or remotely actuated using a magnetic field. The kirigami cut pattern results in a simple unit cell that can be connected together in series, and again in parallel, to create kirigami gripper arrays capable of simultaneously grasping multiple delicate and slippery objects. These soft and lightweight grippers will have applications in robotics, haptics, and biomedical device design.

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. We show that due to a mechanical analogy between pressure and curvature, positive natural curvatures can severely destabilize a thin shell, while negative natural curvatures can strengthen the shell against buckling, providing the possibility to design shells that buckle at or above the theoretical limit for pressure alone, i.e., a strengthening factor. These results extend directly from the classical analysis of the stability of shells under pressure, and highlight the important role that nonmechanical stimuli can have on modifying the membrane state of stress in a thin 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. Experimentally, we vary the arc length of an elastic loop injected into an array of soft, spherical grains at various initial number densities. At low initial packing fractions, the elastic loop deforms as though it were hitting a flat surface by periodically folding into the array. Above a critical packing fraction φc, local re-orientations within the granular medium create an effectively curved surface leading to the emergence of a distinct circular packing morphology. These results bring new insights into the packing behavior of wires and thin sheets, and will be relevant to modeling plant root morphogenesis, burrowing and locomotive strategies of vertebrates & invertebrates, and developing smart, steerable needles.

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. FEM simulations are performed using a reduced model for the swelling process to aid us in rationalizing the underlying mechanism, providing a qualitative agreement with experiments. We believe that the working principle of our mechanism can be extended to bimetallic shells undergoing variations in temperature and to shells made of pH-responsive gels, where the change in knockdown factor could be changed dynamically.

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. We present a model that interprets this scaling as the competition between radial and circumferential bending. Next, we show that reducing the transverse confinement of saddles causes the lobe number to decrease with a similar scaling analysis. Hence, one geometric parameter captures the wave number through a wide range of radial and transverse confinement, connecting the shell shape to the shape of the boundary that confines it. We expect these results to be relevant for an expanse of shell shapes, and thus applicable to the design of shape-shifting materials and the swelling and growth of soft structures.

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. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-through, while also making these shells more 'robust' to snap-through. Our work sheds new light on the robustness of the 'mirror buckling' symmetry of spherical shell caps.

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. However, overconfinement of the elastica will vertically dislodge grains, a form of stress relaxation in the granular medium that illustrates the intricate coupling in elastogranular interactions.

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. The instabilities, and their dependence on geometry, are rationalized by reducing the spontaneous curvature to an effective mechanical load. This formulation reveals a combined pressurelike term in the bulk and a torquelike term in the boundary, allowing scaling predictions for the instabilities that are in excellent agreement with experiments and simulations. Moreover, the effective pressure analogy suggests a curvature-induced subcritical buckling in closed shells. We determine the critical buckling curvature via a linear stability analysis that accounts for the combination of residual membrane and bending stresses. The prominent role of geometry in our findings suggests the applicability of the results over a wide range of scales.

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. We show that the extended lubrication theory is a robust tool for an accurate estimate of pressure drop in channels with shape changes on the order of the channel height, accounting for both smooth and sharp changes in geometry.

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. Our analytical model shows that these deformations are only weakly dependent on thickness, which we confirm with experiments on centimeter-scale objects and molecular dynamics simulations of graphene and MoS2 nanoscale sheets. We show how the interactions between non-propagating cracks can enable either lift or rotation, and we use a combination of experiments, theory, continuum computational analysis, and molecular dynamics simulations to provide mechanistic insights into the geometric and topological design of kirigami actuators.

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. Based on the classical fracture mechanics approach for rate-independent materials, the load train compliance should not affect the force capacity of the adhesive system, whereas when the area to compliance ratio is used as the scaling parameter, it directly influences the bond strength, making it necessary to distinguish compliance contributions. To verify the scaling relationship, single lap shear tests were performed for a given pressure sensitive adhesive (PSA) tape specimens with different bond areas, number of backing layers, and load train compliance. The shear lag model was used to derive closed-form relationships for the system compliance and its derivative with respect to the debond area. Digital image correlation (DIC) is implemented to verify the non-uniform shear stress distribution obtained from the shear lag model in a lap shear geometry. The results obtained from this approach could lead to a better understanding of the relationship between bond strength and the geometry and mechanical properties of adhesive systems.

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. In particular, we rationalize the transition between coalescence-dominated, bending-dominated, and swelling-dominated regimes, and enumerate the subtle interfacial mechanisms at play in the ratcheting of a fluid droplet trapped between the curling beams.

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. We find that the mean curvature of the isometric state is three fourths the natural curvature, which we verify by numerics and experiments. We finally investigate the emergence of a preferred direction of bending in the isometric state, guided by numerical analyses. The scalability of our model suggests that it is suitable to describe the morphing of sheets spanning several orders of magnitude.

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. The morphing of these geometric composites is dictated by both swelling and geometry, with diffusion controlling the swelling-induced actuation, and geometric confinement dictating the structure's deformed shape. Building on a simple mechanical analog, we present an analytical model that quantitatively describes how the Gaussian and mean curvatures of a thin disk are affected by the interplay among geometry, mechanics, and swelling. This model is in excellent agreement with our experiments and numerics. We show that the dynamics of residual swelling is dictated by a competition between two characteristic diffusive length scales governed by geometry. Our results provide the first 2D analog of Timoshenko's classical formula for the thermal bending of bimetallic beams - our generalization explains how the Gaussian curvature of a 2D geometric composite is affected by geometry and elasticity. The understanding conferred by these results suggests that the controlled shaping of geometric composites may provide a simple complement to traditional manufacturing techniques.