Intrinsically anomalous self-similarity of randomly folded matter

Abstract

We found that randomly folded thin sheets exhibit unconventional scale invariance, which we termed as an intrinsically anomalous self-similarity, because the self-similarity of the folded configurations and of the set of folded sheets are characterized by different fractal dimensions. Besides, we found that self-avoidance does not affect the scaling properties of folded patterns, because the self-intersections of sheets with finite bending rigidity are restricted by the finite size of crumpling creases, rather than by the condition of self-avoidance. Accordingly, the local fractal dimension of folding structures is found to be universal (Dl=2.64+/-0.05) and close to expected for a randomly folded phantom sheet with finite bending rigidity. At the same time, self-avoidance is found to play an important role in the scaling properties of the set of randomly folded sheets of different sizes, characterized by the material-dependent global fractal dimension D<Dl. So intrinsically anomalous self-similarity is expected to be an essential feature of randomly folded thin matter.

Entropic rigidity of a crumpling network in a randomly folded thin sheet

“…More recently, it was found that the internal structure of folding configurations also possesses scaling invariance within a wide range of length scales [43]. The fractal dimension of folding configurations is found to be universal 05 .…”

“…The fractal scaling behavior (2) was observed in numerous experiments with randomly folded papers [35,37,43,47,51,52,53,54,55,56], metal foils [18,35,42,57,58], and cream layers [38].…”

“…Accordingly, the folding phenomena are associated with a rich class of crumpling phenomena [3], which belong to a wider class of interfacial deformation phenomena [5]. Because of their biological and technological importance, the properties of randomly folded thin materials are now the subject of increasingly growing attention [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45].…”

“…The properties of ridges have been studied thoroughly. Scaling laws governing the energy and size of the ridge have been obtained analytically [5,49] and tested numerically [9,32,48,50] and experimentally [36,37,43].…”

“…0 64 . 2 ± = l D [43], whereas the fractal dimension D of the set of balls folded from sheets with different size L is the thickness dependent (see ref. [37]).…”