Thin elastic solids are easily deformed into a myriad of three-dimensional shapes, which may contain sharp localized structures as in a crumpled candy wrapper, or have smooth and diffuse features like the undulating edge of a flower. Anticipating and controlling these morphologies is crucial to a variety of applications involving textiles, synthetic skins, and inflatable structures. Here we show that a "wrinkle-to-crumple" transition, previously observed in specific settings, is a ubiquitous response for confined sheets. This unified picture is borne out of a suite of model experiments on polymer films confined to liquid interfaces with spherical, hyperbolic, and cylindrical geometries, which are complemented by experiments on macroscopic membranes inflated with gas. We use measurements across this wide range of geometries, boundary conditions, and lengthscales to quantify several robust morphological features of the crumpled phase, and we build an empirical phase diagram for crumple formation that disentangles the competing effects of curvature and compression. Our results suggest that crumples are a generic microstructure that emerge at large curvatures due to a competition of elastic and substrate energies.
We report an experimental study of the development of orientational order in a crumpled sheet, with a particular focus on the role played by the geometry of confinement. Our experiments are performed on elastomeric sheets immersed in a fluid, so that the effects of plasticity and friction are suppressed. When the sheet is crumpled either axially or radially within a cylinder, we find that the sheet aligns with the flat or the curved wall, depending on the aspect ratio of the cylinder. Nematic correlations develop between the normals of the sheets at relatively low volume fractions and the crum…
Crumpled thin sheets are complex fractal structures whose physical properties are influenced by a hierarchy of ridges. In this Letter, we report experiments that measure the stress-strain relation and show the coexistence of phases in the stretching of crumpled surfaces. The pull stress showed a change from a linear Hookean regime to a sublinear scaling with an exponent of $0.65 \pm 0.03$, which is identified with the Hurst exponent of the crumpled sheets. The stress fluctuations are studied, the statistical distribution of force peaks is analyzed and it is shown how the unpacking of crumpled …
Despite the apparent ease with which a sheet of paper is crumpled and tossed away, crumpling dynamics are often considered a paradigm of complexity. This complexity arises from the infinite number of configurations a disordered crumpled sheet can take. Here we experimentally show that key aspects of crumpling have a very simple description; the evolution of the damage in crumpling dynamics can largely be described by a single global quantity, the total length of all creases. We follow the evolution of the damage network in repetitively crumpled elastoplastic sheets, and show that the dynamics …
As a confined thin sheet crumples, it spontaneously segments into flat facets delimited by a network of ridges. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Experiments have shown that the total crease length accrues logarithmically when repeatedly compacting and unfolding a sheet of paper. Here, we offer insight to this unexpected result by exploring the correspondence between crumpling and fragmentation processes. We identify a physical model for the evolution of facet area and ridge length distributions of crumple…
Hand crumpled paper balls involve intricate structure with a network of creases and vertices, yet show simple scaling properties, which suggests self-similarity of the structure. We investigate the internal structure of crumpled papers by the micro computed tomography (micro-CT) without destroying or unfolding them. From the reconstructed three dimensional data, we examine several power laws for the crumpled square sheets of paper of the sizes $L=50\sim 300$ mm, and obtain the mass fractal dimension $D_M = 2.7\pm 0.1$ by the relation between the mass and the radius of gyration of the balls, an…
When a thin sheet is crushed into a small three-dimensional volume, it invariably forms a structure with a low volume fraction but high resistance to further compression. Being a far-from-equilibrium process, forced crumpling is not necessarily amenable to a statistical description in which the parameters of the initially flat sheet and the final confinement fully specify the resulting crumpled state. Instead, the internal geometry and mechanical properties of the crumpled ball may reflect the history of its preparation. Our X-ray microtomography experiments reveal that the internal 3-dimensio…
We introduce a toy model of crumpled sheets. We use simulation to show there is a first order phase transition in the model, from a disordered dilute phase to a mixture with a layered phase.
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 is 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 ca…
We study whether and how the energy scalings based on the single-ridge approximation are revised in an actual crumpled sheet; namely, in the presence of ridge-ridge interactions. Molecular Dynamics Simulation is employed for this purpose. In order to improve the data quality, modifications are introduced to the common protocol. As crumpling proceeds, we find that the average storing energy changes from being proportional to one-third of the ridge length to a linear relation, while the ratio of bending and stretching energies decreases from 5 to 2. The discrepancy between previous simulations a…
We examine the crescent singularity of a developable cone in a setting similar to that studied by Cerda et al [Nature 401, 46 (1999)]. Stretching is localized in a core region near the pushing tip and bending dominates the outer region. Two types of stresses in the outer region are identified and shown to scale differently with the distance to the tip. Energies of the d-cone are estimated and the conditions for the scaling of core region size R_c are discussed. Tests of the pushing force equation and direct geometrical measurements provide numerical evidence that core size scales as R_c ~ h^{1…
A three-dimensional X-ray tomography is performed to investigate the internal structure and its evolution of a crumpled aluminum foil. The upper and lower bounds of the internal geometric fractal dimension are determined, which increase with the compression. Contrary to the simulation results, we find that the mass distribution changes from being inhomogeneous to uniform. Corroborated with the evidence from previous experiments, these findings support the physical picture that the elastic property precedes the plastic one at dominating the deformation and mechanical response for all crumpled s…
Crumpling an ordinary thin sheet transforms it into a structure with unusual mechanical behaviors, such as enhanced rigidity, emission of crackling noise, slow relaxations, and memory retention. A central challenge in explaining these behaviours lies in understanding the contribution of the complex geometry of the sheet. Here, we combine cyclic driving protocols and 3D imaging to correlate the global mechanical response and the underlying geometric transformations in unfolded crumpled sheets. We find that their response to cyclic strain is intermittent, hysteretic, and encodes a memory of the …
We investigate the propagation of transverse elastic waves in crumpled media. We set up the wave equation for transverse waves on a generic curved, strained surface via a Langrangian formalism and use this to study the scaling behaviour of the dispersion curves near the ridges and on the flat facets. This analysis suggests that ridges act as barriers to wave propagation and that modes in a certain frequency regime could be trapped in the facets. A simulation study of the wave propagation qualitatively supported our analysis and showed interesting effects of the ridges on wave propagation.
X-ray tomography is performed to acquire 3D images of crumpled aluminum foils. We develop an algorithm to trace out the labyrinthian paths in the three perpendicular cross sections of the data matrices. The tangent-tangent correlation function along each path is found to decay exponentially with an effective persistence length that shortens as the crumpled ball becomes more compact. In the mean time, we observed ordered domains near the crust, similar to the lamellae phase mixed by the amorphous portion in lyotropic liquid crystals. The size and density of these domains grow with further compa…
Crumpled sheets have a surprisingly large resistance to further compression. We have studied the crumpling of thin sheets of Mylar under different loading conditions. When placed under a fixed compressive force, the size of a crumpled material decreases logarithmically in time for periods up to three weeks. We also find hysteretic behavior when measuring the compression as a function of applied force. By using a pre-treating protocol, we control this hysteresis and find reproducible scaling behavior for the size of the crumpled material as a function of the applied force.
The network formed by ridges in a straightened sheet of crumpled paper is studied using a laser profilometer. Square sheets of paper were crumpled into balls, unfolded and their height profile measured. From these profiles the imposed ridges were extracted as networks. Nodes were defined as intersections between ridges, and links as the various ridges connecting the nodes. Many network and spatial properties have been investigated. The tail of the ridge length distribution was found to follow a power-law whereas the shorter ridges followed a log-normal distribution. The degree distribution was…
The physics of tightly packed structures of a wire and other threadlike materials confined in cavities has been explored in recent years in connection with crumpled systems and a number of topics ranging from applications to DNA packing in viral capsids and surgical interventions with catheter to analogies with the electron gas at finite temperature and with theories of two-dimensional quantum gravity. When a long piece of wire is injected into two-dimensional cavities, it bends and originates in the jammed limit a series of closed structures that we call loops. In this work we study the extra…
We measure the geometry of a crumpled sheet of paper with laser-aided topography and discuss its statistical properties. The curvature of an elasto-plastic fold scales linearly with applied force. The curvature distribution follows an exponential form with regions of high curvature localized along ridges. The measured ridge length distribution is consistent with a hierarchical model for ridge breaking during crumpling. A large fraction of the ridges are observed to terminate without bifurcating and the ridge network connectedness is not as complete as anticipated. The self-affinity of the surf…
We investigate the formation dynamics of plastic creases in thin elasto-plastic sheets. In contrast to the commonly accepted description of crumpled thin sheets, which asserts that creases form only by elastic interaction between two d-cones, the creases we study in this letter are created by plastic deformations left in the wake of a single propagating d-cone. Upon application of load, a d-cone initially remains stationary and responds by deforming globally. However, above a critical load, the d-cone undergoes a sharpening transition that focuses the stresses at its tip, allowing it to propag…
We investigate the dissipative mechanisms exhibited by creased material sheets when subjected to mechanical loading, which comes in the form of plasticity and relaxation phenomena within the creases. After demonstrating that plasticity mostly affects the rest angle of the creases, we devise a mapping between this quantity and the macroscopic state of the system that allows us to track its reference configuration along an arbitrary loading path, resulting in a powerful monitoring and design tool for crease-based metamaterials. Furthermore, we show that complex relaxation phenomena, in particula…
A growing or compressed thin elastic sheet adhered to a rigid substrate can exhibit a buckling instability, forming an inward hump. Our study shows that the strip morphology depends on the delicate balance between the compression energy and the bending energy. We find that this instability is a first order phase transition between the adhered solution and the buckled solution whose main control parameter is related to the sheet stretchability. In the nearly- unstretchable regime we provide an analytic expression for the critical threshold. Compressibility is the key assumption which allows us …
We study the behavior of thin elastic sheets that are bent and strained under the influence of weak, smooth confinement. We show that the emerging shapes exhibit the coexistence of two types of domains that differ in their characteristic stress distributions and energies, and reflect different constraints. A focused-stress patch is subject to a geometric, piecewise-inextensibility constraint, whereas a diffuse-stress region is characterized by a mechanical constraint - the dominance of a single component of the stress tensor. We discuss the implications of our findings for the analysis of elas…
The nonlinear response of driven complex materials -- disordered magnets, amorphous media, crumpled sheets -- features intricate transition pathways where the system repeatedly hops between metastable states. % which encode memory effects. Such pathways encode memory effects and may allow information processing -- yet tools are lacking to experimentally observe and control these pathways, and their full breadth has not been explored. Here we introduce compression of corrugated elastic sheets to precisely observe and manipulate {their full}, multi-step pathways, which are reproducible, robust,a…
We examine the shape change of a thin disk with an inserted wedge of material when it is pushed against a plane, using analytical, numerical and experimental methods. Such sheets occur in packaging, surgery and nanotechnology. We approximate the sheet as having vanishing strain, so that it takes a conical form in which straight generators converge to a disclination singularity. Then its shape is that which minimizes elastic bending energy alone. Real sheets are expected to approach this limiting shape as their thickness approaches zero. The planar constraint forces a sector of the sheet to buc…
Since the discovery of graphene oxide (GO), the most accessible of the precursors of graphene, this material has been widely studied for applications in science and technology. In this work, we describe a procedure to obtain GO dispersions in water at high concentrations, these highly dehydrated dispersions being in addition fully redispersible by dilution. With the availability of such concentrated samples, it was possible to investigate the structure of hydrated GO sheets in a previously unexplored range of concentrations, and to evidence a structural phase transition. Tentatively applying m…
A thin fiber or sheet curled into a circular container has a detached region whose shape and force ratios are independent of material properties and container radius. We compute this shape and compare it with experiments. The discrete forces acting at either end of the detached region have a ratio that depends only on the length of the fiber or sheet relative to the circle radius. We calculate this force ratio in three regimes of circle radius.
Hydrodynamic forces may significantly affect the motion of polymers. In sheet-like cavities, such as the cell's cytoplasm and microfluidic channels, the hydrodynamic forces are long-range. It is therefore expected that that hydrodynamic interactions will dominate the motion of polymers in sheets and will be manifested by Zimm-like scaling. Quite the opposite, we note here that although the hydrodynamic forces are long-range their overall effect on the motion of polymers vanishes due to the symmetry of the two-dimensional flow. As a result, the predicted scaling of experimental observables such…
We study the deformation of a fluctuating crystalline sheet confined between two flat rigid walls as a simple model for layered solids where bonds among atoms {\it within} the same layer are much stronger than those {\it between} layers. When subjected to sufficiently high loads in an appropriate geometry, these solids deform and fail in unconventional ways. Recent experiments suggest that configurations named {\it ripplocations}, where a layer folds backwards over itself, are involved. These structures are distinct and separated by large free energy barriers from smooth {\it ripples} of the a…
The phase diagram of Yukawa particles confined between two parallel hard walls is calculated at zero-temperature beyond the bilayer regime by lattice-sum-minimization. Tuning the screening, a rich phase behavior is found in the regime bounded by stable two-triangular layers and 3-square layers. In this regime, alternating prism phases with square and triangular basis, structures derived from a hcp bulk lattice, and a structure with two outer layers and two inner staggered rectangular layers, reminiscent of a Belgian waffle iron, are stable. These structures are verifiable in experiments on cha…
Casimir force in a water nanolayer confined by graphene sheets
Zernike Institute for Advanced Materials, University of Groningen（格罗宁根大学泽尼克高级材料研究所）；Chinese Academy of Sciences Key Laboratory of Soft Matter Physics, Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所软物质物理重点实验室）；Wenzhou Institute, University of Chinese Academy of Sciences（中国科学院大学温州研究院）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）
First principles methods based on periodic boundary conditions are used extensively by materials theorists. However, applying these methods to systems with confined electronic states entails the use of large unit cells in order to avoid artificial image interactions. We present a general approach for truncating the Coulomb interaction that removes image effects directly and leads to well converged results for modest-sized periodic cells. As an illustration, we find the lowest-energy quasiparticle and exciton states in two-dimensional hexagonal GaN sheets. These sheets have been proposed as par…
Crumpled paper or drapery patterns are everyday examples of how elastic sheets can respond to external forcing. In this Letter, we study experimentally a novel 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 taco-like 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 de…
Spatially confined rigid membranes reorganize their morphology in response to the imposed constraints. A crumpled elastic sheet presents a complex pattern of random folds focusing the deformation energy while compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy. Here, we study the energy distribution for highly confined membranes and show the emergence of a new morphological instability triggered by a period-doubling bifurcation. A periodic self-organized focalization of the deformation energy is observed provi…
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 wit…
Bacteria often form surface-bound communities, embedded in a self-produced extracellular matrix, called biofilms. Quantitative studies of their growth have typically focused on unconfined expansion above solid or semi-solid surfaces, leading to exponential radial growth. This geometry does not accurately reflect the natural or biomedical contexts in which biofilms grow in confined spaces. Here we consider one of the simplest confined geometries: a biofilm growing laterally in the space between a solid surface and an overlying elastic sheet. A poroelastic framework is utilised to derive the rad…
