Effect of ridge-ridge interactions in crumpled thin sheets

Abstract

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 and experiments on the material-dependence for the power-law exponent is resolved. We further determine the averaged ridge length to scale linearly with the crumpled ball size R, the ridge number as 1/R 2 , and the average storing energy per unit ridge length as 1/R 2.364∼2.487 . These results are consistent with the mean-field predictions. Finally, we extend the existent simulations to the high-pressure region for completeness, and verify the existence of a new scaling relation that is more general than the familiar power law at covering the whole density range. 46.32.+x, 89.75.Fb Although crumpling is ubiquitous and simple to enact, intense researches to understand its complexities only began in the last thirty years or so. It has not only become relevant to cutting-edge technologies like utilizing crumpled graphene sheets[1] to harvest energies by converting motion into electricity, but is of interest to the general phenomenon of condensation of many outstanding problems in physics [2]. However, some properties of crumpling remain unresolved. For instance, as the developable cones[3-6] increase in number and form the familiar network[7-9] of ridges and vertices, scientists still do not know how they collaborate to produce stunningly simple power laws between the crumpled ball size R and crumpling force F [10-19], R ∼ F −α and for the occurring frequency [20][21][22] of different noise intensities. One challenge is to understand how the crumpled sheet constructs spontaneously a highly porous and yet robust structure [23][24][25]. Proper theoretical tools are also in demand to tackle the complex many-body interactions among these deformations, especially the selfavoidance that plays an important role [14] at creating the glass-like interior after a series of highly non-equilibrium processes [26], similar to the random packing in a golf ball basket or salt jar.In view of the theoretic inability, Molecular Dynamics Simulation becomes a powerful tool to obtain information such as the three-dimensional distribution of ridges and facets before unfolding, energy storage in each ridge, and how the increase of crumpled ball density affects these quantities. We are interested in the effect of ridgeridge interactions on the energy scaling predicted by the single-ridge approximation [2,7,9]. Attentions are also paid to resolve major disagreements between previous simulations [14,18] and experiments [16,19]. In the mean time, we pu...

Fold analysis of crumpled sheet using micro computed tomography

“…The estimated value of the Hurst exponent H ≈ 0.9 for the short length scale suggests that the line configuration is quite ballistic for the length scale up to R g , but it eventually approaches the random walk for the longer scale. The value of the Hurst exponent H ≈ 0.9 is consistent with the relation (22) to the exponent for the structure factor β line ≈ 1.1, but not with the relation (24) to α line ≈ 0.74 or the mass fractal dimension D M ≈ 2.7. In other words, for the line on a crumpled paper, the self-similarity of each line structure in the short length scale is not consistent with the overall scaling upon changing the size L in contrast to the case of the whole structure of a crumpled paper, in which case D M = d f , thus the self-similarity of the internal structure is consistent with the global scaling.…”

“…The internal structure of the crumpled paper ball has been studied by examining the crease networks on an unfolded sheets [15][16][17][18], the cross sections obtained by cutting the balls in half [11,18,19], or the sequences of holes made by a needle piercing through the balls [13] as well as numerical simulations [12,[20][21][22]. These are indirect way of observing the internal structure.…”

Crumple-Origami Transition for Twisting Cylindrical Shells

“…In the initial stage of crumpling, the kite model by Witten [19] can predict the ratio of stretching and bending energies on each ridge and how their sum increases with the ridge length. As ridge-ridge interactions become important, how the resistence force, average length and number of ridges vary with the radius of crumpled ball have also been deduced by theory [20], molecular dynamics simulation and experiments [12,20,21].…”