Comparative Study of Crumpling and Folding of Thin Sheets

Abstract

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 6 or 7 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 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.

A Crumpled Sheet’s Remembrance of Things Past

“…The value for maximum compression H 2 ¼ 5 mm for the crumpled sheet and H 2 ¼ 3 mm for the elastic foams was chosen such that the experiments remain in the linear strain-stress regime. For higher compressions, we observed a transition in to a power-law dependence [21,[30][31][32]. Under these conditions, we observe reproducible logarithmic relaxation curves, as shown in Fig.…”

Furrows in the wake of propagating d-cones

“…An example for this may be found in the mesmerizing network of scars that decorate the face of a sheet of paper after it has been crumpled to a ball and then opened 1 . In recent years, crumpled paper has attracted much attention in the scientific community and has been investigated analytically 2,3 , numerically 4,5 and experimentally [6][7][8][9] .…”

Mechanical Response of a Creased Sheet

“…6 shows the two opposite responses of a simple accordion like origami to an applied loading when the size of the panels is modified. Using simple arguments [8], it is shown that the energy stored in the crease, and consequently the mechanical parameter κ, scales as B/h, which implies that the characteristic length scale L * is a linear function of the thickness. Fig.…”

“…This simple protocol leads to reproducible results and, in turn, governs the mechanical properties of the hinge. This feature can be understood qualitatively by noticing that the creation of the crease involves localized plastic deformations, through localized storage of bending elastic energy [8]. Assuming an ideally plastic behavior of the material, and considering the thickness h as the crease characteristic radius of curvature, the plastic strain scales as p ∼ hφ p /h = φ p , where φ p is the characteristic angular region in which the irreversible deformations are localized.…”

“…In defiance of these enchanting associations, the prototypical system which allows a more controlled study of folds creation and of their mechanical properties assumes a far more undistinguished form: namely a crumpled paper [1,8]. Careful observations have revealed that decoration of a thin plate by a crease network dramatically modifies the global mechanical response of the plate resulting, for instance, in aging of the structure [9], negative Poisson ratio and unusual response to bending and stretching [10].…”