Morphological phases of crumpled wire

Indentation of an elastic arch on a frictional substrate: Pinning, unfolding, and snapping

In this study, we investigate the morphology and mechanics of a naturally curved elastic arch loaded at its center and frictionally supported at both ends on a flat, rigid substrate. Through systematic numerical simulations, we classify the observed behaviors of the arch into three configurations in terms of the arch geometry and the coefficient of static friction with the substrate. A linear theory is developed based on a planar elastica model combined with Amontons-Coulomb's frictional law, which quantitatively explains the numerically constructed phase diagram. The snapping transition of a loaded arch in a sufficiently large indentation regime, which involves a discontinuous force jump, is numerically observed. The proposed model problem enables a fully analytical investigation and demonstrates a rich variety of mechanical behaviors owing to the interplay among elasticity, geometry, and friction. This study provides a basis for understanding more common but complex systems, such as a cylindrical shell subjected to a concentrated load and simultaneously supported by frictional contact with surrounding objects.

Packing of elastic rings with friction

We study the deformations of elastic filaments confined within slowly shrinking circular boundaries, under contact forces with friction. We perform computations with a spring-lattice model that deforms like a thin inextensible filament of uniform bending stiffness. Early in the deformation, two lobes of the filament make contact. If the friction coefficient is small enough, one lobe slides inside the other; otherwise, the lobes move together or one lobe bifurcates the other. There follows a sequence of deformations that is a mixture of spiralling and bifurcations, primarily the former with small friction and the latter with large friction. With zero friction, a simple model predicts that the maximum curvature and the total elastic energy scale as the wall radius to the − 3 / 2 and − 2 powers, respectively. With non-zero friction, the elastic energy follows a similar scaling but with a prefactor up to eight times larger, due to delayering and bending with a range of small curvatures. For friction coefficients as large as 1, the deformations are qualitatively similar with and without friction at the outer wall. Above 1, the wall friction case becomes dominated by buckling near the wall.

Asymptotic softness of a laterally confined sheet in a pressurized chamber

Elastohydrodynamic models, that describe the interaction between a thin sheet and a fluid medium, have been proven successful in explaining the complex behavior of biological systems and artificial materials. Motivated by these applications we study the quasistatic deformation of a thin sheet that is confined between the two sides of a closed chamber. The two parts of the chamber, above and below the sheet, are filled with an ideal gas. We show that the system is governed by two dimensionless parameters, Δ and η, that account respectively for the lateral compression of the sheet and the ratio between the amount of fluid filling each part of the chamber and the bending stiffness of the sheet. When η≪1 the bending energy of the sheet dominates the system, the pressure drop between the two sides of the chamber increases, and the sheet exhibits a symmetric configuration. When η≫1 the energy of the fluid dominates the system, the pressure drop vanishes, and the sheet exhibits an asymmetric configuration. The transition between these two limiting scenarios is governed by a third branch of solutions that is characterized by a rapid decrease of the pressure drop. Notably, across the transition the energetic gap between the symmetric and asymmetric states scales as δE∼Δ^{2}. Therefore, in the limit Δ≪1 small variations in the energy are accompanied by relatively large changes in the elastic shape.

Inversion of force lines in fiber-reinforced jammed granular material

Freestanding columns, built out of nothing but loose gravel and continuous strings can be stable even at several meters in height and withstand vertical loads high enough to severely fragment grains of the column core. We explain this counter-intuitive behavior through dynamic simulations with polyhedral rigid particles and elastic wire chains. We evaluate the fine structure of the particle contact networks, as well as confining forces and reveal fundamental intrinsic differences to the well-studied case of confining silos.

Mechanics of two filaments in tight orthogonal contact

Networks of flexible filaments often involve regions of tight contact. Predictively understanding the equilibrium configurations of these systems is challenging due to intricate couplings between topology, geometry, large nonlinear deformations, and friction. Here, we perform an in-depth study of a simple, yet canonical, problem that captures the essence of contact between filaments. In the orthogonal clasp, two filaments are brought into contact, with each centerline lying in one of a pair of orthogonal planes. Our data from X-ray tomography (μCT) and mechanical testing experiments are in excellent agreement with finite element method (FEM) simulations. Despite the apparent simplicity of the physical system, the data exhibit strikingly unintuitive behavior, even when the contact is frictionless. Specifically, we observe a curvilinear diamond-shaped ridge in the contact-pressure field between the two filaments, sometimes with an inner gap. When a relative displacement is imposed between the filaments, friction is activated, and a highly asymmetric pressure field develops. These findings contrast to the classic capstan analysis of a single filament wrapped around a rigid body. Both the μCT and FEM data indicate that the cross-sections of the filaments can deform significantly. Nonetheless, an idealized geometrical theory assuming undeformable tube cross-sections and neglecting elasticity rationalizes our observations qualitatively and highlights the central role of the small, but nonzero, tube radius of the filaments. We believe that our orthogonal clasp analysis provides a building block for future modeling efforts in frictional contact mechanics of more complex filamentary structures.

Morphological Transition between Patterns Formed by Threads of Magnetic Beads

Magnetic beads attract each other, forming chains. We push such chains into an inclined Hele-Shaw cell and discover that they spontaneously form self-similar patterns. Depending on the angle of inclination of the cell, two completely different situations emerge; namely, above the static friction angle the patterns resemble the stacking of a rope and below they look similar to a fortress from above. Moreover, locally the first pattern forms a square lattice, while the second pattern exhibits triangular symmetry. For both patterns, the size distributions of enclosed areas follow power laws. We characterize the morphological transition between the two patterns experimentally and numerically and explain the change in polarization as a competition between friction-induced buckling and gravity.

Patterns formed by chains of magnetic beads

Magnetic beads attract each other forming rather stable chains. We consider such chains formed by magnetic beads and push them into a Hele-Shaw cell either from the boundary or from the center. When such a chain is pushed into a cavity, it bends and folds spontaneously forming interesting unreported patterns. These patterns are self-similar and an effective fractal dimension can be defined. As found experimentally and with numerical simulations, the numbers of beads, loops and contacts follow power laws as a function of packing fraction and, depending on the injection procedure, even energetically less favorable triangular configurations can be stabilized.

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.

Compaction of quasi-one-dimensional elastoplastic materials

Insight into crumpling or compaction of one-dimensional objects is important for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here, we report how friction, plasticity and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size and injection force.

Principles underlying crumpling of one-dimensional objects may be relevant to both biomolecular processes and to design of mechanical devices. By compacting various wires under rigid confinement and modelling observed geometric features, the authors show how friction, plasticity and torsion enhance disorder and lead to a transition from coiled to folded geometries.

Slip Morphology of Elastic Strips on Frictional Rigid Substrates

The morphology of an elastic strip subject to vertical compressive stress on a frictional rigid substrate is investigated by a combination of theory and experiment. We find a rich variety of morphologies, which-when the bending elasticity dominates over the effect of gravity-are classified into three distinct types of states: pinned, partially slipped, and completely slipped, depending on the magnitude of the vertical strain and the coefficient of static friction. We develop a theory of elastica under mixed clamped-hinged boundary conditions combined with the Coulomb-Amontons friction law and find excellent quantitative agreement with simulations and controlled physical experiments. We also discuss the effect of gravity in order to bridge the difference in the qualitative behaviors of stiff strips and flexible strings or ropes. Our study thus complements recent work on elastic rope coiling and takes a significant step towards establishing a unified understanding of how a thin elastic object interacts vertically with a solid surface.

Packing loops into annular cavities

The continuous packing of a flexible rod in two-dimensional cavities yields a countable set of interacting domains that resembles nonequilibrium cellular systems and belongs to a new class of lightweight material. However, the link between the length of the rod and the number of domains requires investigation, especially in the case of non-simply connected cavities, where the number of avoided regions emulates an effective topological temperature. In the present article we report the results of an experiment of injection of a single flexible rod into annular cavities in order to find the total length needed to insert a given number of loops (domains of one vertex). Using an exponential model to describe the experimental data we quite minutely analyze the initial conditions, the intermediary behavior, and the tight packing limit. This method allows the observation of a new fluctuation phenomenon associated with instabilities in the dynamic evolution of the packing process. Furthermore, the fractal dimension of the global pattern enters the discussion under a novel point of view. A comparison with the classical problems of the random close packing of disks and jammed disk packings is made.

Scaling, crumpled wires, and genome packing in virions

The packing of a genome in virions is a topic of intense current interest in biology and biological physics. The area is dominated by allometric scaling relations that connect, e.g., the length of the encapsulated genome and the size of the corresponding virion capsid. Here we report scaling laws obtained from extensive experiments of packing of a macroscopic wire within rigid three-dimensional spherical and nonspherical cavities that can shed light on the details of the genome packing in virions. We show that these results obtained with crumpled wires are comparable to those from a large compilation of biological data from several classes of virions.

JS, Gomes MAF, Herrmann HJ. Crumpling Damaged Graphene

Through molecular mechanics we find that non-covalent interactions modify the fractality of crumpled damaged graphene. Pristine graphene membranes are damaged by adding random vacancies and carbon-hydrogen bonds. Crumpled membranes exhibit a fractal dimension of 2.71 ± 0.02 when all interactions between carbon atoms are considered, and 2.30 ± 0.05 when non-covalent interactions are suppressed. The transition between these two values, obtained by switching on/off the non-covalent interactions of equilibrium configurations, is shown to be reversible and independent on thermalisation. In order to explain this transition, we propose a theoretical model that is compatible with our numerical findings. Finally, we also compare damaged graphene membranes with other crumpled structures, as for instance polymerised membranes and paper sheets, that share similar scaling properties.

Unpacking of a Crumpled Wire from Two-Dimensional Cavities

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 extraction of a crumpled tightly packed wire from a circular cavity aiming to remove loops individually. The size of each removed loop, the maximum value of the force needed to unpack each loop, and the total length of the extracted wire were measured and related to an exponential growth and a mean field model consistent with the literature of crumpled wires. Scaling laws for this process are reported and the relationship between the processes of packing and unpacking of wire is commented upon.

Wrinkles and folds in a fluid-supported sheet of finite size

A laterally confined thin elastic sheet lying on a liquid substrate displays regular undulations, called wrinkles, characterized by a spatially extended energy distribution and a well-defined wavelength λ. As the confinement increases, the deformation energy is progressively localized into a single narrow fold. An exact solution for the deformation of an infinite sheet was previously found, indicating that wrinkles in an infinite sheet are unstable against localization for arbitrarily small confinement. We present an extension of the theory to sheets of finite length L, accounting for the experimentally observed wrinkle-to-fold transition. We derive an exact solution for the periodic deformation in the wrinkled state, and an approximate solution for the localized, folded state. We find that a second-order transition between these two states occurs at a critical confinement Δ(F)=λ(2)/L.

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.

Crumpled states of a wire in a cubic cavity with periodic obstacles

In this paper, we study experimentally the configurations of a plastic wire injected into a cubic cavity containing periodic obstacles placed along a fixed direction. The wire moves in a wormlike manner within the cavity until it becomes jammed in a crumpled state. The maximum packing fraction of the wire depends on the topology of the cavity, which in turn depends on the number of obstacles. The experimental results exhibit scaling laws and display similarities as well as differences with a recently reported two-dimensional version of this complex packing problem. We discuss in detail several aspects of this problem that seem as intricate as the problem of a self-avoiding random walk. Analogies between the experiment reported and some statistical aspects of the bond-percolation problem, as well as of the interacting electron gas at finite temperature, and other physical issues are also discussed.

Comparative study of crumpling and folding of thin sheets

Crumpling and folding of paper are at first sight very different ways of confining thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly inefficient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than six or seven times. Here we show that the stiffness that builds up in the two processes is of the same nature, and therefore simple folding models allow us to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We find that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.

Ordered packing of elastic wires in a sphere

In this paper we study the ordered packing of wires in a sphere. We propose an analytical model and compare the model predictions with the results of our experiments and simulations for the maximum packing fraction, the number of formed coils, the fractal dimension, and bending energy. We show that the relative system size [i.e., the ratio of the wire radius to the sphere radius (a/R)] is the most important control parameter for the maximum packing fraction. We find that the number of coils obeys a power-law relation of the form N∼(R/a){1.5} and the fractal dimension of the structures is 2.5, independent of the system size. Our theoretical results are in good agreement with the experimental data and the predictions of the numerical simulations.

Packing of elastic wires in spherical cavities

We investigate the morphologies and maximum packing density of thin wires packed into spherical cavities. Using simulations and experiments, we find that ordered as well as disordered structures emerge, depending on the amount of internal torsion. We find that the highest packing densities are achieved in low torsion packings for large systems, but in high torsion packings for small systems. An analysis of both situations is given in terms of energetics and comparison is made to analytical models of DNA packing in viral capsids.

Mechanical response of a self-avoiding membrane: fold collisions and the birth of conical singularities

An elastic membrane that is forced to reside in a container smaller than its natural size will deform and upon further volume reduction eventually crumple. The crumpled state is characterized by the localization of energy in a complex network of highly deformed crescent-like regions joined by line ridges. In this article we study through a combination of experiments, numerical simulations, and analytic approaches the emergence of localized regions of high stretching when a self-avoiding membrane is subject to a severe geometrical constraint. Based on our experimental observations and numerical results we suggest that at moderate packing fraction interlayer interactions produce a response equivalent to that of a thicker membrane that has the shape of the deformed one. We find that new conical dislocations, coined satellite d-cones, appear as the deformed membrane further compactifies. When these satellite d-cones are born, a substantial relaxation of the mechanical response of the membrane is observed. Evidence is found that friction plays a key role in stabilizing the folded structures.

Self-contact and instabilities in the anisotropic growth of elastic membranes

We investigate the morphology of thin discs and rings growing in the circumferential direction. Recent analytical results suggest that this growth produces symmetric excess cones (e cones). We study the stability of such solutions considering self-contact and bending stress. We show that, contrary to what was assumed in previous analytical solutions, beyond a critical growth factor, no symmetric e cone solution is energetically minimal any more. Instead, we obtain skewed e cone solutions having lower energy, characterized by a skewness angle and repetitive spiral winding with increasing growth. These results are generalized to discs with varying thickness and rings with holes of different radii.

Fractal topology of hand-crumpled paper

We study the statistical topology of folding configurations of hand folded paper balls. Specifically, we are studying the distribution of two sides of the sheet along the ball surface and the distribution of sheet fragments when the ball is cut in half. We found that patterns obtained by mapping of ball surface into unfolded flat sheet exhibit the fractal properties characterized by two fractal dimensions which are independent on the sheet size and the ball diameter. The mosaic patterns obtained by sheet reconstruction from fragments of two parts (painted in two different colors) of the ball cut in half also possess a fractal scale invariance characterized by the box fractal dimension DBF=1.68 ± 0.04 , which is independent on the sheet size. Furthermore, we noted that DBF, at least numerically, coincide with the universal fractal dimension of the intersection of hand folded paper ball with a plane. Some other fractal properties of folding configurations are recognized.

Crumpled states of a wire in a two-dimensional cavity with pins

In this paper, we report an extensive experimental study of the configurations of a plastic wire injected into a two-dimensional planar cavity populated with fixed pins. The wire is not allowed to cross any pin, but it can move in a wormlike manner within the cavity until to become jammed in a crumpled state. The jammed packing fraction depends heavily on the topology of the cavity, which depends on the number of pins. The experiment reveals nontrivial entanglement effects and scaling laws which are largely independent of the details of the distribution of pins, the symmetry of the cavity or the type of the wire. A mean-field model for the process is presented and analogies with some basic aspects of statistical thermodynamics are discussed.

X-ray tomography of a crumpled plastoelastic thin sheet

Three-dimensional x-ray tomography is performed to investigate the internal structure and its evolution of a crumpled aluminum foil ball. 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 structures. We show that the interior of a crumpled ball at the plastic regime can be mapped to the compact packing of a granular system.

Structure and dynamics of vibrated granular chains: comparison to equilibrium polymers

We show that the statistical properties of a vibrated granular bead chain are similar to standard models of polymers in equilibrium. Granular chains of length up to N=1024 beads were confined within a circular vibrating bed, and their configurations were imaged. To differentiate the effects of persistence and confinement on the chain, we compared with simulations of both persistent random-walk (RW) and self-avoiding walk (SAW) models. Static properties, such as the radius of gyration and structure factor, are governed for short chains (N<or=128) by persistence and can be matched by those of RWs. Self-avoidance and confinement effects are both important for longer chains and the results are well described by equilibrated SAWs. We also find that the collective dynamics of the granular chain is similar to the Rouse model of polymers. In particular, as long as confinement is negligible, the center of mass of the chain diffuses with a diffusion coefficient that scales as 1/N, and the dynamic structure factor decays exponentially in time.

Crumpling wires in two dimensions

An energy-minimal simulation is proposed to study the patterns and mechanical properties of elastically crumpled wires in two dimensions. Our aim is to describe the behavior at the intermediate occupancy of the cavity so that its radius of gyration is varied up to one twentieth of the wire length. We tuned the bending rigidity and stretching modulus to measure the energy allocation, size-mass exponent, and the stiffness exponent. The mass exponent is shown to be constant at value D_{M}=1.33 , so is the stiffness exponent alpha=-0.25 . But the latter varies with the plasticity parameters s and theta_{p} . These numerical findings agree excellently with the experimental results.