Fractal features of a crumpling network in randomly folded thin matter and mechanics of sheet crushing

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.

Fold analysis of crumpled sheets using microcomputed tomography

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 microcomputed tomography (micro-CT) without destroying or unfolding them. From the reconstructed three-dimensional (3D) data, we examine several power laws for the crumpled square sheets of paper of the sizes L=50-300 mm and obtain the mass fractal dimension D_{M}=2.7±0.1 by the relation between the mass and the radius of gyration of the balls and the fractal dimension 2.5≲d_{f}≲2.8 for the internal structure of each crumpled paper ball by the box counting method in the real space and the structure factors in the Fourier space. The data for the paper sheets are consistent with D_{M}=d_{f}, suggesting that the self-similarity in the structure of each crumpled ball gives rise to the similarity among the balls with different sizes. We also examine the cellophane sheets and the aluminium foils of the size L=200 mm and obtain 2.6≲d_{f}≲2.8 for both of them. The micro-CT also allows us to reconstruct 3D structure of a line drawn on the crumpled sheets of paper. The Hurst exponent for the root-mean-square displacement along the line is estimated as H≈0.9 for the length scale shorter than the scale of the radius of gyration, beyond which the line structure becomes more random with H∼0.5.

A model for the fragmentation kinetics of crumpled thin sheets

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 crumpled sheets, and propose a mechanism for re-fragmentation driven by geometric frustration. This mechanism establishes a feedback loop in which the facet size distribution informs the subsequent rate of fragmentation under repeated confinement, thereby producing a new size distribution. We then demonstrate the capacity of this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon.

Crumpling of thin sheets as a basis for creating mechanical metamaterials

Crumpled thin sheets exhibit extraordinary characteristics such as a high strength combined with a low volume ratio. This review focuses on the physics of crumpled thin sheets, including the crumpling mechanics, crumpling methods, and the mechanical behavior of crumpled thin sheets. Most of the physical and mechanical properties of crumpled thin sheets change with the compaction ratio, which creates the opportunity to obtain the properties that are needed for a specific application simply by changing the compaction ratio. This also enables obtaining unusual combinations of material properties, which cannot be easily found in nature. Furthermore, crumpling starts from a flat surface, which could first be decorated with (nano-) patterns or functionalized through other surface treatment techniques, many of which are only applicable to flat surfaces. Ultimately, the crumpling of thin sheets could be used for creating disordered mechanical metamaterials, which are less sensitive to geometric imperfections compared to ordered designs of mechanical metamaterials that are based, for example, on origami or lattice structures.

Random formation of layers and ridges through the crumpling of a flat matter can form a robust mechanical metamaterial.

Crumpling-based soft metamaterials: the effects of sheet pore size and porosity

Crumpled-based materials are relatively easy to fabricate and show robust mechanical properties for practical applications, including meta-biomaterials design aimed for improved tissue regeneration. For such requests, however, the structure needs to be porous. We introduce a crumpled holey thin sheet as a robust bio-metamaterial and measure the mechanical response of a crumpled holey thin Mylar sheet as a function of the hole size and hole area fraction. We also study the formation of patterns of crease lines and ridges. The area fraction largely dominated the crumpling mechanism. We also show, the crumpling exponents slightly increases with increasing the hole area fraction and the total perimeter of the holes. Finally, hole edges were found to limit and guide the propagation of crease lines and ridges.

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.

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.

Effective degrees of freedom of a random walk on a fractal

We argue that a non-Markovian random walk on a fractal can be treated as a Markovian process in a fractional dimensional space with a suitable metric. This allows us to define the fractional dimensional space allied to the fractal as the ν-dimensional space F(ν) equipped with the metric induced by the fractal topology. The relation between the number of effective spatial degrees of freedom of walkers on the fractal (ν) and fractal dimensionalities is deduced. The intrinsic time of random walk in F(ν) is inferred. The Laplacian operator in F(ν) is constructed. This allows us to map physical problems on fractals into the corresponding problems in F(ν). In this way, essential features of physics on fractals are revealed. Particularly, subdiffusion on path-connected fractals is elucidated. The Coulomb potential of a point charge on a fractal embedded in the Euclidean space is derived. Intriguing attributes of some types of fractals are highlighted.

Edwards's statistical mechanics of crumpling networks in crushed self-avoiding sheets with finite bending rigidity

This paper is devoted to the crumpling of thin matter. The Edwards-like statistical mechanics of crumpling networks in a crushed self-avoiding sheet with finite bending rigidity is developed. The statistical distribution of crease lengths is derived. The relationship between sheet packing density and hydrostatic pressure is established. The entropic contribution to the crumpling network rigidity is outlined. The effects of plastic deformations and sheet self-contacts on crumpling mechanics are discussed. Theoretical predictions are in good agreement with available experimental data and results of numerical simulations. Thus, the findings of this work provide further insight into the physics of crumpling and mechanical properties of crumpled soft matter.

Effect of ridge-ridge interactions in crumpled thin sheets

We study whether and how the energy scaling based on the single-ridge approximation is 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 and experiments on the material-dependence for the power-law exponent is resolved. We further determine the average ridge length to scale as 1/D(1/3), the ridge number as D(2/3), and the average storing energy per unit ridge length as D(0.881) where D denotes the volume density of the crumpled ball. These results are accompanied by experimental proofs and are consistent with mean-field predictions. Finally, we extend the existent simulations to the high-pressure region and verify the existence of a scaling relation that is more general than the familiar power law at covering the whole density range.

Statistics of energy dissipation and stress relaxation in a crumpling network of randomly folded aluminum foils

We study stress relaxation in hand folded aluminum foils subjected to the uniaxial compression force F(λ). We found that once the compression ratio is fixed (λ=const) the compression force decreases in time as F∝F_{0}P(t), where P(t) is the survival probability time distribution belonging to the domain of attraction of max-stable distribution of the Fréchet type. This finding provides a general physical picture of energy dissipation in the crumpling network of a crushed elastoplastic foil. The difference between energy dissipation statistics in crushed viscoelastic papers and elastoplastic foils is outlined. Specifically, we argue that the dissipation of elastic energy in crushed aluminum foils is ruled by a multiplicative Poisson process governed by the maximum waiting time distribution. The mapping of this process into the problem of transient random walk on a fractal crumpling network is suggested.