Buckling mediated by mobile localized elastic excitations

Experiments reveal that structural transitions in thin sheets are mediated by the passage of transient and stable mobile localized elastic excitations. These "crumples" or "d-cones" nucleate, propagate, interact, annihilate, and escape. Much of the dynamics occurs on millisecond time scales. Nucleation sites correspond to regions where generators of the ideal unstretched surface converge. Additional stable intermediate states illustrate two forms of quasistatic inter-crumple interaction through ridges or valleys. These interactions create pairs from which extended patterns may be constructed in larger specimens. The onset of localized transient deformation with increasing sheet size is correlated with a characteristic stable crumple size, whose measured scaling with thickness is consistent with prior theory and experiment for localized elastic features in thin sheets. We offer a new theoretical justification of this scaling.

Plastic instability of annular crystalline membrane in circular confinement

Understanding the mechanical instabilities of two-dimensional membranes has strong connection to the subjects of structure instabilities, morphology control, and materials failures. In this work, we investigate the plastic mechanism developed in the annular crystalline membrane system for adapting to the shrinking space, which is caused by the controllable gradual expansion of the inner boundary. In the process of plastic deformation, we find the continuous generation of dislocations at the inner boundary and their collective migration to the outer boundary; this neat dynamic scenario of dislocation current captures the complicated reorganization process of the particles. We also reveal the characteristic vortex structure arising from the interplay of topological defects and the displacement field. These results may find applications in the precise control of structural instabilities in packings of particulate matter and covalently bonded systems.

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.

Curved crease origami and topological singularities at a cellular scale enable hyper-extensibility of Lacrymaria olor

Eukaryotic cells undergo dramatic morphological changes during cell division, phagocytosis and motility. Fundamental limits of cellular morphodynamics such as how fast or how much cellular shapes can change without harm to a living cell remain poorly understood. Here we describe hyper-extensibility in the single-celled protist Lacrymaria olor, a 40 μm cell which is capable of reversible and repeatable extensions (neck-like protrusions) up to 1500 μm in 30 seconds. We discover that a unique and intricate organization of cortical cytoskeleton and membrane enables these hyper-extensions that can be described as the first cellular scale curved crease origami. Furthermore, we show how these topological singularities including d-cones and twisted domain walls provide a geometrical control mechanism for the deployment of membrane and microtubule sheets as they repeatably spool thousands of time from the cell body. We lastly build physical origami models to understand how these topological singularities provide a mechanism for the cell to control the hyper-extensile deployable structure. This new geometrical motif where a cell employs curved crease origami to perform a physiological function has wide ranging implications in understanding cellular morphodynamics and direct applications in deployable micro-robotics.

Dynamics of fluctuating thin sheets under random forcing

We study the dynamic structure factor of fluctuating elastic thin sheets subject to conservative (athermal) random forcing. In Steinbock et al. [Phys. Rev. Res. 4, 033096 (2022)2643-156410.1103/PhysRevResearch.4.033096] the static structure factor of such a sheet was studied. In this paper we recap the model developed there and investigate its dynamic properties. Using the self-consistent expansion, the time-dependent two-point function of the height profile is determined and found to decay exponentially in time. Despite strong nonlinear coupling, the decay rate of the dynamic structure factor is found to coincide with the effective coupling constant for the static properties, which suggests that the model under investigation exhibits certain quasilinear behavior. Confirmation of these results by numerical simulations is also presented.

Indentation of an elastic disk on a circular supporting ring

Thin elastic two-dimensional systems under compressive stresses may relieve part of their stretching energy by developing out-of-plane undulations. We investigate experimentally and theoretically the indentation of an elastic disk supported by a circular ring and show that compressive stresses are relieved via two different routes: either developing buckles which are spread over the system or developing a d-cone where deformation is concentrated in a subregion of the system. We characterize the indentation threshold for wrinkles or d-cone existence as a function of aspect ratio.

Shape reconfiguration through origami folding sets an upper limit on drag

Mechanisms of drag reduction through shape reconfiguration have been extensively studied on model geometries of plates and beams that deform primarily in bending. Adding an origami crease pattern to such plates produces a distinct class of deformation modes, with large shape changes along selected degrees of freedom. Here, we investigate the impact of those creases on reconfiguration processes and on drag, focusing on the waterbomb base as a generic case. When placed in a uniform airflow, this origami unit folds into a compact structure, whose frontal area collapses with increasing flow velocity. It enhances drag reduction to the point that fluid loading eventually ceases to increase with flow speed, reaching an upper limit. We further show that this limit is adjustable through the origami structural parameters: the stiffness and rest angle of the folds, and their pattern. Experimental results, corroborated by a fluid–elastic theoretical model, point to a scenario consistent with the previous literature: reconfiguration is governed by a dimensionless Cauchy number that measures the competition between fluid loading and elastic resistance to deformation, here embodied in creases. This foldable system yet stands out through the rare passive drag-capping lever it provides, a valuable asset for self-protection in strong wind.

The Föppl-von Kármán equations of elastic plates with initial stress

Initially, stressed plates are widely used in modern fabrication techniques, such as additive manufacturing and UV lithography, for their tunable morphology by application of external stimuli. In this work, we propose a formal asymptotic derivation of the Föppl-von Kármán equations for an elastic plate with initial stresses, using the constitutive theory of nonlinear elastic solids with initial stresses under the assumptions of incompressibility and material isotropy. Compared to existing works, our approach allows us to determine the morphological transitions of the elastic plate without prescribing the underlying target metric of the unstressed state of the elastic body. We explicitly solve the derived FvK equations in some physical problems of engineering interest, discussing how the initial stress distribution drives the emergence of spontaneous curvatures within the deformed plate. The proposed mathematical framework can be used to tailor shape on demand, with applications in several engineering fields ranging from soft robotics to four-dimensional printing.

Tensional twist-folding of sheets into multilayered scrolled yarns

Twisting sheets as a strategy to form functional yarns relies on millennia of human practice in making catguts and fabric wearables, but it still lacks overarching principles to guide their intricate architectures. We show that twisted hyperelastic sheets form multilayered self-scrolled yarns, through recursive folding and twist localization, that can be reconfigured and redeployed. We combine weakly nonlinear elasticity and origami to explain the observed ordered progression beyond the realm of perturbative models. Incorporating dominant stretching modes with folding kinematics, we explain the measured torque and energetics originating from geometric nonlinearities due to large displacements. Complementarily, we show that the resulting structures can be algorithmically generated using Schläfli symbols for star-shaped polygons. A geometric model is then introduced to explain the formation and structure of self-scrolled yarns. Our tensional twist-folding framework shows that origami can be harnessed to understand the transformation of stretchable sheets into self-assembled architectures with a simple twist.

Finite curved creases in infinite isometric sheets

Geometric stress focusing, e.g., in a crumpled sheet, creates pointlike vertices that terminate in a characteristic local crescent shape. The observed scaling of the size of this crescent is an open question in the stress focusing of elastic thin sheets. According to experiments and simulations, this size depends on the outer dimension of the sheet, but intuition and rudimentary energy balance indicate it should only depend on the sheet thickness. We address this discrepancy by modeling the observed crescent with a more geometric approach, where we treat the crescent as a curved crease in an isometric sheet. Although curved creases have already been studied extensively, the crescent in a crumpled sheet has its own unique features: the material crescent terminates within the material, and the material extent is indefinitely larger than the extent of the crescent. These features together with the general constraints of isometry lead to constraints linking the surface profile to the crease-line geometry. We construct several examples obeying these constraints, showing finite curved creases are fully realizable. This approach has some particular advantages over previous analyses, as we are able to describe the entire material without having to exclude the region around the sharp crescent. Finally, we deduce testable relations between the crease and the surrounding sheet and discuss some of the implications of our approach with regards to the scaling of the crescent size.

Thermalized buckling of isotropically compressed thin sheets

The buckling of thin elastic sheets is a classic mechanical instability that occurs over a wide range of scales. In the extreme limit of atomically thin membranes like graphene, thermal fluctuations can dramatically modify such mechanical instabilities. We investigate here the delicate interplay of boundary conditions, nonlinear mechanics, and thermal fluctuations in controlling buckling of confined thin sheets under isotropic compression. We identify two inequivalent mechanical ensembles based on the boundaries at constant strain (isometric) or at constant stress (isotensional) conditions. Remarkably, in the isometric ensemble, boundary conditions induce a novel long-ranged nonlinear interaction between the local tilt of the surface at distant points. This interaction combined with a spontaneously generated thermal tension leads to a renormalization group description of two distinct universality classes for thermalized buckling, realizing a mechanical variant of Fisher-renormalized critical exponents. We formulate a complete scaling theory of buckling as an unusual phase transition with a size-dependent critical point, and we discuss experimental ramifications for the mechanical manipulation of ultrathin nanomaterials.

Buckling-Fracture Transition and the Geometrical Charge of a Crack

We present a unifying approach that describes both surface bending and fracture in the same geometrical framework. An immediate outcome of this view is a prediction for a new mechanical transition: the buckling-fracture transition. Using responsive gel strips that are subjected to nonuniform osmotic stress, we show the existence of the transition: Thin plates do not fracture. Instead, they release energy via buckling, even at strains that can be orders of magnitude larger than the Griffith fracture criterion. The analysis of the system reveals the dependence of the transition on system's parameters and agrees well with experimental results. Finally, we suggest a new description of a mode I crack as a line distribution of Gaussian curvature. It is thus exchangeable with extrinsic generation of curvature via buckling. The work opens the way for the study of mechanical problems within a single nonlinear framework. It suggests that fracture driven by internal stresses can be completely avoided by a proper geometrical design.

Transition to stress focusing for locally curved sheets

A rectangular thin elastic sheet is deformed by forcing a contact between two points at the middle of its length. A transition to buckling with stress focusing is reported for the sheets sufficiently narrow with a critical width proportional to the sheet length with an exponent 2/3 in the small thickness limit. Additionally, a spring network model is solved to explore the thick sheet limit and to validate the scaling behavior of the transition in the thin sheet limit. The numerical results reveal that buckling does not exist for the thickest sheets, and a stability criterion is established for the buckling of a curved sheet.

Nature's forms are frilly, flexible, and functional

A ubiquitous motif in nature is the self-similar hierarchical buckling of a thin lamina near its margins. This is seen in leaves, flowers, fungi, corals, and marine invertebrates. We investigate this morphology from the perspective of non-Euclidean plate theory. We identify a novel type of defect, a branch-point of the normal map, that allows for the generation of such complex wrinkling patterns in thin elastic hyperbolic surfaces, even in the absence of stretching. We argue that branch points are the natural defects in hyperbolic sheets, they carry a topological charge which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating elastic energy. We develop a theory for branch points and investigate their role in determining the mechanical response of hyperbolic sheets to weak external forces.

More views of a one-sided surface: mechanical models and stereo vision techniques for Möbius strips

Möbius strips are prototypical examples of ribbon-like structures. Inspecting their shapes and features provides useful insights into the rich mechanics of elastic ribbons. Despite their ubiquity and ease of construction, quantitative experimental measurements of the three-dimensional shapes of Möbius strips are surprisingly non-existent in the literature. We propose two novel stereo vision-based techniques to this end—a marker-based technique that determines a Lagrangian description for the construction of a Möbius strip, and a structured light illumination technique that furnishes an Eulerian description of its shape. Our measurements enable a critical evaluation of the predictive capabilities of mechanical theories proposed to model Möbius strips. We experimentally validate, seemingly for the first time, the developable strip and the Cosserat plate theories for predicting shapes of Möbius strips. Equally significantly, we confirm unambiguous deficiencies in modelling Möbius strips as Kirchhoff rods with slender cross-sections. The experimental techniques proposed and the Cosserat plate model promise to be useful tools for investigating a general class of problems in ribbon mechanics.

Ridge energy for thin nematic polymer networks

Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).

Buckling transitions and soft-phase invasion of two-component icosahedral shells

What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component "invades" the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.

Foldable cones as a framework for nonrigid origami

The study of origami-based mechanical metamaterials usually focuses on the kinematics of deployable structures made of an assembly of rigid flat plates connected by hinges. When the elastic response of each panel is taken into account, novel behaviors take place, as in the case of foldable cones (f-cones): circular sheets decorated by radial creases around which they can fold. These structures exhibit bistability, in the sense that they can snap through from one metastable configuration to another. In this work, we study the elastic behavior of isometric f-cones for any deflection and crease mechanics, which introduce nonlinear corrections to a linear model studied previously. Furthermore, we test the inextensibility hypothesis by means of a continuous numerical model that includes both the extended nature of the creases, stretching and bending deformations of the panels. The results show that this phase-field-like model could become an efficient numerical tool for the study of realistic origami structures.

Instabilities in dielectric elastomers: buckling, wrinkling, and crumpling

Instabilities in a thin sheet are ubiquitous and can be induced by various stimuli, such as a uniaxial force, liquid-vapor surface tension, etc. This paper investigates voltage-induced instabilities in a membrane of a dielectric elastomer. Instabilities including buckling, wrinkling, and crumpling are observed in the experiments. The prestretches of the dielectric elastomer are found to play a significant role in determining its instability mode. When the prestretch is small, intermediate, or large, the membrane may undergo buckling, wrinkling, or crumpling, respectively. Finite element analysis is conducted to study these instability modes, and the simulations are well consistent with the experimental observations. We hope that this investigation of mechanical and physical properties of dielectric elastomers can enhance their extensive and significant applications in soft devices and soft robots.

The compressive strength of crumpled matter

Crumpling a sheet creates a unique, stiff and lightweight structure. Use of crumples in engineering design is limited because there are not simple, physically motivated structure-property relations available for crumpled materials; one cannot trust a crumple. On the contrary, we demonstrate that an empirical model reliably predicts the reaction of a crumpled sheet to a compressive force. Experiments show that the prediction is quantitative over 50 orders of magnitude in force, for purely elastic and highly plastic polymer films. Our data does not match recent theoretical predictions based on the dominance of building-block structures (bends, folds, d-cones, and ridges). However, by directly measuring substructures, we show clearly that the bending in the stretching ridge is responsible for the strength of both elastic and plastic crumples. Our simple, predictive model may open the door to the engineering use of a vast range of materials in this state of crumpled matter.

Crumpled matter hasn’t been widely used to solve real world engineering problems largely due to the lack of quantitative models. Croll et al. show that it is the bending in ridges making both elastic and plastic sheets resistant to compression and describe the mechanical response using an empirical model.

Machine learning in a data-limited regime: Augmenting experiments with synthetic data uncovers order in crumpled sheets

Machine learning has gained widespread attention as a powerful tool to identify structure in complex, high-dimensional data. However, these techniques are ostensibly inapplicable for experimental systems where data are scarce or expensive to obtain. Here, we introduce a strategy to resolve this impasse by augmenting the experimental dataset with synthetically generated data of a much simpler sister system. Specifically, we study spontaneously emerging local order in crease networks of crumpled thin sheets, a paradigmatic example of spatial complexity, and show that machine learning techniques can be effective even in a data-limited regime. This is achieved by augmenting the scarce experimental dataset with inexhaustible amounts of simulated data of rigid flat-folded sheets, which are simple to simulate and share common statistical properties. This considerably improves the predictive power in a test problem of pattern completion and demonstrates the usefulness of machine learning in bench-top experiments where data are good but scarce.

Extreme contractility and torsional compliance of soft ribbons under high twist

We investigate experimentally and model the mechanical response of a soft Hookean ribbon submitted to large twist η and longitudinal tension T, under clamped boundary conditions. We derive a formula for the torque M using the Föppl-von Kàrmàn equations up to third order in twist, incorporating a twist-tension coupling. In the stable helicoid regime, quantitative agreement with experimental data is obtained. When twisted above a critical twist η_{L}(T), ribbons develop wrinkles and folds which modify qualitatively the mechanical behavior. We show a surprisingly large longitudinal contraction upon twist, reminiscent of a Poynting effect, and a much lower torsional stiffness. Far from threshold, we identify two regimes depending on the applied T. In a high-T regime, we find that the torque scales as ηT and the contraction as η^{2}, in agreement with a far from threshold analysis where compression and bending stresses are neglected. In a low-T regime, the contraction still scales as η^{2} but the torque appears T independent and linear with η. We argue that the large curvature of the folds now contributes significantly to the torque. This regime is discussed in the context of asymptotic isometry for very thin plates submitted to vanishing tension but large change of shape, as in crumpling.

Tailoring relaxation dynamics and mechanical memory of crumpled materials by friction and ductility

Crumpled sheets show slow mechanical relaxation and long lasting memory of previous mechanical states. By using uniaxial compression tests, the role of friction and ductility on the stress relaxation dynamics of crumpled systems is investigated. We find a material dependent relaxation constant that can be tuned by changing ductility and adhesive properties of the sheet. After a two-step compression protocol, nonmonotonic aging is reported for polymeric, elastomeric and metal sheets, with relaxation dynamics that are dependent on the material's properties. These findings can contribute to tailoring and programming of crumpled materials to get desirable mechanical properties.

Local mechanical description of an elastic fold

To go beyond the simple model for the fold as two flexible surfaces or faces linked by a crease that behaves as an elastic hinge, we carefully shape and anneal a crease within a polymer sheet and study its mechanical response. First, we carry out an experimental study that involves recording both the shape of the fold in various loading configurations and the associated force needed to deform it. Then, an elastic model of the fold is built upon a continuous description of both the faces and the crease as a thin sheet with a non-flat reference configuration. The comparison between the model and experiments yields the local fold properties and explains the significant differences we observe between tensile and compression regimes. Furthermore, an asymptotic study of the fold deformation enables us to determine the local shape of the crease and identify the origin of its mechanical behaviour.

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.

Kirigami Mechanics as Stress Relief by Elastic Charges

We develop a geometric approach to understand the mechanics of perforated thin elastic sheets, using the method of strain-dependent image elastic charges. This technique recognizes the buckling response of a hole under an external load as a geometrically tuned mechanism of stress relief. We use a diagonally pulled square paper frame as a model system to quantitatively test and validate our approach. Specifically, we compare nonlinear force-extension curves and global displacement fields in theory and experiment. We find a strong softening of the force response accompanied by curvature localization at the inner corners of the buckled frame. Counterintuitively, though in complete agreement with our theory, for a range of intermediate hole sizes, wider frames are found to buckle more easily than narrower ones. Upon extending these ideas to many holes, we demonstrate that interacting elastic image charges can provide a useful kirigami design principle to selectively relax stresses in elastic materials.

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.

Wrinkling patterns in soft shells

Curvature plays an important role in the morphological evolution of soft shells under stretch. Here, through a combination of experiment, theory and simulation, we investigate the behavior of a hemispherical soft shell subject to an increasing outward point force at its pole. In contrast to an inward point force inducing a polygonal pattern of buckling in the shell, we observe a four-stage morphological transition and symmetry breaking under an increasing outward point force. The shell undergoes axisymmetric deformation around its pole and then buckles into a non-axisymmetric shape with a number of shallow wrinkles emanating from the pole, followed by the emergence of crater-like deep crumples and ultimately a transformation into a wrinkled pseudocone. Our theoretical analysis and numerical simulations yield the critical conditions for the morphological transitions at each stage of deformation and reveal the underlying interplays between elastic bending and stretching energies and the curvature of the shell.

Intermittency and emergence of coherent structures in wave turbulence of a vibrating plate

We report numerical investigations of wave turbulence in a vibrating plate. The possibility to implement advanced measurement techniques and long-time numerical simulations makes this system extremely valuable for wave turbulence studies. The purely 2D character of dynamics of the elastic plate makes it much simpler to handle compared to much more complex 3D physical systems that are typical of geo- and astrophysical issues (ocean surface or internal waves, magnetized plasmas or strongly rotating and/or stratified flows). When the forcing is small the observed wave turbulence is consistent with the predictions of the weak turbulent theory. Here we focus on the case of stronger forcing for which coherent structures can be observed. These structures look similar to the folds and D-cones that are commonly observed for strongly deformed static thin elastic sheets (crumpled paper) except that they evolve dynamically in our forced system. We describe their evolution and show that their emergence is associated with statistical intermittency (lack of self similarity) of strongly nonlinear wave turbulence. This behavior is reminiscent of intermittency in Navier-Stokes turbulence. Experimental data show hints of the weak to strong turbulence transition. However, due to technical limitations and dissipation, the strong nonlinear regime remains out of reach of experiments and therefore has been explored numerically.

The smectic order of wrinkles

A thin elastic sheet lying on a soft substrate develops wrinkled patterns when subject to an external forcing or as a result of geometric incompatibility. Thin sheet elasticity and substrate response equip such wrinkles with a global preferred wrinkle spacing length and with resistance to wrinkle curvature. These features are responsible for the liquid crystalline smectic-like behaviour of such systems at intermediate length scales. This insight allows better understanding of the wrinkling patterns seen in such systems, with which we explain pattern breaking into domains, the properties of domain walls and wrinkle undulation. We compare our predictions with numerical simulations and with experimental observations.

A thin elastic sheet can develop wrinkles which arrange into patterns similar to those characteristic of liquid crystals. Here the authors use this analogy to propose a mapping between the elastic sheet problem and the smectic liquid crystal problem which can enable a better understanding of wrinkling.

Effect of the material properties on the crumpling of a thin sheet

While simple at first glance, the dense packing of sheets is a complex phenomenon that depends on material parameters and the packing protocol. We study the effect of plasticity on the crumpling of sheets of different materials by performing isotropic compaction experiments on sheets of different sizes and elasto-plastic properties. First, we quantify the material properties using a dimensionless foldability index. Then, the compaction force required to crumple a sheet into a ball as well as the average number of layers inside the ball are measured. For each material, both quantities exhibit a power-law dependence on the diameter of the crumpled ball. We experimentally establish the power-law exponents and find that both depend nonlinearly on the foldability index. However the exponents that characterize the mechanical response and morphology of the crumpled materials are related linearly. A simple scaling argument explains this in terms of the buckling of the sheets, and recovers the relation between the crumpling force and the morphology of the crumpled structure. Our results suggest a new approach to tailor the mechanical response of the crumpled objects by carefully selecting their material properties.

Inverted cones and their elastic creases

We study the elastic inversion of a right circular cone, in particular, the uniform shape of the narrow crease that divides its upright and inverted parts. Our methodology considers a cylindrical shell analogy for simplicity where the crease is the boundary layer deformation. Solution of its governing equation of deformation requires careful crafting of the underlying assumptions and boundary conditions in order to reveal an expression for the crease shape in closed form. We can then define the characteristic width of crease exactly, which is compared to a geometrically nonlinear, large displacement finite element analysis. This width is shown to be accurately predicted for shallow and steep cones, which imparts confidence to our original assumptions. Using the shape of crease, we compute the strain energy stored in the inverted cone, in order to derive an expression for the applied force of inversion by a simple energy method. Again, our predictions match finite element data very well. This study may complement other studies of creases traditionally formed in a less controlled manner, for example, during crumpling of lightweight sheets.

k-cones and kirigami metamaterials

We are inspired by the tensile buckling of a thin sheet with a slit to create a foldable planar metamaterial. The buckled shape comprises two pairs of identical e-cones connected to the slit, which we refer to as a k-cone. We approximate this shape as discrete vertices that can be folded out of plane as the slit is pulled apart. We determine their kinematics and we calculate generic shape properties using a simple elastic model of the folded shape. We then show how the folded sheet may be tessellated as a unit cell within a larger sheet, which may be constructed a priori by cutting and folding the latter in a regular way, in order to form a planar kirigami structure with a single degree of freedom.

From damage to discovery via virtual unwrapping: Reading the scroll from En-Gedi

Computer imaging techniques are commonly used to preserve and share readable manuscripts, but capturing writing locked away in ancient, deteriorated documents poses an entirely different challenge. This software pipeline-referred to as "virtual unwrapping"-allows textual artifacts to be read completely and noninvasively. The systematic digital analysis of the extremely fragile En-Gedi scroll (the oldest Pentateuchal scroll in Hebrew outside of the Dead Sea Scrolls) reveals the writing hidden on its untouchable, disintegrating sheets. Our approach for recovering substantial ink-based text from a damaged object results in readable columns at such high quality that serious critical textual analysis can occur. Hence, this work creates a new pathway for subsequent textual discoveries buried within the confines of damaged materials.

Fundamental conical defects: The d-cone, its e-cone, and its p-cone

We consider well-known surface disclinations by cutting, joining, and folding pieces of paper card. The resulting shapes have a discrete, folded vertex whose geometry is described easily by Gauss's mapping, in particular, we can relate the degree of angular excess, or deficit, to the size of fold line rotations by the area enclosed by the vector diagram of these rotations. This is well known for the case of a so-called "d-cone" of zero angular deficit, and we formulate the same for a general disclination. This method allows us to observe kinematic properties in a meaningful way without needing to consider equilibrium. Importantly, the simple vector nature of our analysis shows that some disclinations are primitive; and that other types, such as d-cones, are amalgamations of them.

Disclinations, e-cones, and their interactions in extensible sheets

We investigate the nucleation, growth, and spatial organization of topological defects with a ribbon shaped elastic sheet which is stretched and twisted. Singularities are found to spontaneously arrange in a triangular lattice in the form of vertices connected by stretched ridges that result in a self-rigidified structure. The vertices are shown to be negative disclinations or e-cones which occur in sheets with negative Gaussian curvature, in contrast with d-cones in sheets with zero-Gaussian curvature. We find the growth of the wrinkled width of the ribbon to be consistent with a far-from-threshold approach assuming a compression-free base state. The system is found to show a transition from a regime where the wavelength is given by the ribbon geometry, to where it is given by its elasticity as a function of the ratio of the applied tension to the elastic modulus and cross-sectional area of the ribbon.

Generic Bistability in Creased Conical Surfaces

The emerging field of mechanical metamaterials has sought inspiration in the ancient art of origami as archetypal deployable structures that carry geometric rigidity, exhibit exotic material properties, and are potentially scalable. A promising venue to introduce functionality consists in coupling the elasticity of the sheet and the kinematics of the folds. In this spirit, we introduce a scale-free, analytical description of a very general class of snap-through, bistable patterns of creases naturally occurring at the vertices of real origami that can be used as building blocks to program and actuate the overall shape of the decorated sheet. These switches appear at the simplest possible level of creasing and admit straightforward experimental realizations.

Indentation of ultrathin elastic films and the emergence of asymptotic isometry

We study the indentation of a thin elastic film floating at the surface of a liquid. We focus on the onset of radial wrinkles at a threshold indentation depth and the evolution of the wrinkle pattern as indentation progresses far beyond this threshold. Comparison between experiments on thin polymer films and theoretical calculations shows that the system very quickly reaches the far from threshold regime, in which wrinkles lead to the relaxation of azimuthal compression. Furthermore, when the indentation depth is sufficiently large that the wrinkles cover most of the film, we recognize a novel mechanical response in which the work of indentation is transmitted almost solely to the liquid, rather than to the floating film. We attribute this unique response to a nontrivial isometry attained by the deformed film, and we discuss the scaling laws and the relevance of similar isometries to other systems in which a confined sheet is subjected to weak tensile loads.

Sheet on a deformable sphere: wrinkle patterns suppress curvature-induced delamination

The adhesion of a stiff film onto a curved substrate often generates elastic stresses in the film that eventually give rise to its delamination. Here we predict that delamination of very thin films can be dramatically suppressed through tiny, smooth deformations of the substrate, dubbed here "wrinklogami," that barely affect the macro-scale topography. This "prolamination" effect reflects a surprising capability of smooth wrinkles to suppress compression in elastic films even when spherical or other doubly curved topography is imposed, in a similar fashion to origami folds that enable construction of curved structures from an unstretchable paper. We show that the emergence of a wrinklogami pattern signals a nontrivial isometry of the sheet to its planar, undeformed state, in the doubly asymptotic limit of small thickness and weak tensile load exerted by the adhesive substrate. We explain how such an "asymptotic isometry" concept broadens the standard usage of isometries for describing the response of elastic sheets to geometric constraints and mechanical loads.

How a curved elastic strip opens

An elastic strip is transversely clamped in a curved frame. The induced curvature decreases as the strip opens and connects to its flat natural shape. Various ribbon profiles are measured and the scaling law for the opening length validates a description where the in-plane stretching gradually relaxes the bending stress. An analytical model of the strip profile is proposed and a quantitative agreement is found with both experiments and simulations of the plates equations. This result provides a unique illustration of smooth nondevelopable solutions in thin sheets.

Morphogenesis of filaments growing in flexible confinements

Space-saving design is a requirement that is encountered in biological systems and the development of modern technological devices alike. Many living organisms dynamically pack their polymer chains, filaments or membranes inside deformable vesicles or soft tissue-like cell walls, chorions and buds. Surprisingly little is known about morphogenesis due to growth in flexible confinements--perhaps owing to the daunting complexity lying in the nonlinear feedback between packed material and expandable cavity. Here we show by experiments and simulations how geometric and material properties lead to a plethora of morphologies when elastic filaments are growing far beyond the equilibrium size of a flexible thin sheet they are confined in. Depending on friction, sheet flexibility and thickness, we identify four distinct morphological phases emerging from bifurcation and present the corresponding phase diagram. Four order parameters quantifying the transitions between these phases are proposed.

Mechanical collapse of confined fluid membrane vesicles

Compact cylindrical and spherical invaginations are common structural motifs found in cellular and developmental biology. To understand the basic physical mechanisms that produce and maintain such structures, we present here a simple model of vesicles in confinement, in which mechanical equilibrium configurations are computed by energy minimization, balancing the effects of curvature elasticity, contact of the membrane with itself and the confining geometry, and adhesion. For cylindrical confinement, the shape equations are solved both analytically and numerically by finite element analysis. For spherical confinement, axisymmetric configurations are obtained numerically. We find that the geometry of invaginations is controlled by a dimensionless ratio of the adhesion strength to the bending energy of an equal area spherical vesicle. Larger adhesion produces more concentrated curvatures, which are mainly localized to the "neck" region where the invagination breaks away from its confining container. Under spherical confinement, axisymmetric invaginations are approximately spherical. For extreme confinement, multiple invaginations may form, bifurcating along multiple equilibrium branches. The results of the model are useful for understanding the physical mechanisms controlling the structure of lipid membranes of cells and their organelles, and developing tissue membranes.

Reversibility of crumpling on compressed thin sheets: reversibility of crumpling

Compressing thin sheets usually yields the formation of singularities which focus curvature and stretching on points or lines. In particular, following the common experience of crumpled paper where a paper sheet is crushed in a paper ball, one might guess that elastic singularities should be the rule beyond some compression level. In contrast, we show here that, somewhat surprisingly, compressing a sheet between cylinders make singularities spontaneously disappear at large compression. This "stress defocusing" phenomenon is qualitatively explained from scale-invariance and further linked to a criterion based on a balance between stretching and curvature energies on defocused states. This criterion is made quantitative using the scalings relevant to sheet elasticity and compared to experiment. These results are synthesized in a phase diagram completed with plastic transitions and buckling saturation. They provide a renewed vision of elastic singularities as a thermodynamic condensed phase where stress is focused, in competition with a regular diluted phase where stress is defocused. The physical differences between phases is emphasized by determining experimentally the mechanical response when stress is focused or defocused and by recovering the corresponding scaling laws. In this phase diagram, different compression routes may be followed by constraining differently the two principal curvatures of a sheet. As evidenced here, this may provide an efficient way of compressing a sheet that avoids the occurrence of plastic damages by inducing a spontaneous regularization of geometry and stress.

Tuning the ordered states of folded rods by isotropic confinement

The packing of elastic objects is increasingly studied in the framework of out-of-equilibrium statistical mechanics and thus these appear to be similar to glassy systems. Here, we present a two-dimensional experiment whereby a rod is confined by a parabolic potential. The setup enables spanning a wide range of folded configurations of the rod. Measurements of the distributions of length and curvature in the system reveal the importance of a stacking process whereby many layers of the rod are grouped into branches. The geometrical order of patterns increases with the confinement strength. Measurements of the distributions of energies lead to the definition of an energy scale that is correlated with the elastic energy of the stacked parts of the rod. This scale imposes energy partition in the system and might be relevant to the framework of the thermodynamics of disordered systems. Following these observations, we describe the patterns as excited states of a ground state corresponding to the most ordered geometry. Eventually, we provide evidence that the disordered state of a folded rod becomes spontaneously closer to the ground state as confinement is increased.

Dipoles in thin sheets

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

Flow-induced draping

Crumpled paper or drapery patterns are everyday examples of how elastic sheets can respond to external forcing. In this Letter, we study experimentally a different sort of forcing. We consider a circular flexible plate clamped at its center and subject to a uniform flow normal to its initial surface. As the flow velocity is gradually increased, the plate exhibits a rich variety of bending deformations: from a cylindrical shape, to isometric developable cones with azimuthal periodicity two or three, to eventually a rolled-up period-three cone. We show that this sequence of flow-induced deformations can be qualitatively predicted by a linear analysis based on the balance between elastic energy and pressure force work.