Topological crossovers in the forced folding of self-avoiding matter

Abstract

We study the scaling properties of forced folding of thin materials of different geometry. The scaling relations implying the topological crossovers from the folding of three-dimensional plates to the folding of two-dimensional sheets and further to the packing of one-dimensional strings are derived for elastic and plastic manifolds. These topological crossovers in the folding of plastic manifolds were observed in experiments with predominantly plastic aluminum strips of different geometry. Elasto-plastic materials, such as paper sheets during the (fast) folding under increasing confinement force, are expected to obey the scaling force-diameter relation derived for elastic manifolds. However, in experiments with paper strips of different geometry we observed the crossover from packing of one-dimensional strings to folding two dimensional sheets only, because the fractal dimension of the set of folded elasto-plastic sheets is the thickness dependent due to the strain relaxation after a confinement force is withdrawn.

Slow dynamics of stress and strain relaxation in randomly crumpled elasto-plastic sheets

“…Accordingly, under increasing hydrostatic pressure, the diameter (R) of a ball, folded from a thin sheet, decreases as R ∝ P −α , where the scaling exponent α is expected to be universal for elastic sheets [26], whereas, in the cases of elasto-plastic and plastic sheets, the value of α is dependent on the energy dissipation in crumpling creases [18,27,28]. Besides, at a fixed pressure (P = const), the diameter of a ball folded from an elasto-plastic or predominantly plastic sheet decreases logarithmically in time for periods of up to several hours [18].…”

“…Furthermore, the force relaxation (5) was studied only in axially compressed predominantly plastic aluminum foils [41], the mechanical behavior of which differs from that of balls folded from elasto-plastic sheets, such as a paper (see Refs. [17,28]). The relaxation of compressive force in specimens folded from crumpled papers has not been tested.…”

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

“…where k and Y2 are the effective bending and stretching moduli of sheet [6], Both moduli of an elastic sheet are dependent on the sheet thickness h [45,46], whereas the apparent stretching modulus of an elastoplastic sheet is thickness independent [34,47], Specifically, the effective bending and stretching moduli of elastic and elastoplastic sheets are equal to respectively, where y3 is the three-dimensional Young modulus and p is the Poisson ratio of the sheet material [46,47].…”

“…However, as the packing density increases, the entropic contribution to the sheet resistance to hydrostatic pressure can become relevant. It is a straightforward matter to derive the relation between the sheet packing density p = L2h/V and hydrostatic pressure (34) E .N Accordingly, using Eqs. (9), (28), (32), and (33), and taking into account that asymptotically P -> oo as p 1, the pressure-packing density relationship can be presented in the following form:…”

“…The crumpling behavior of thin sheets is strongly dependent on the confinement mode [15,22] and sheet geometry [22,34], One of the basic modes is the crumpling of a thin sheet subjected to isotropic confinement [22,[29][30][31]35], Under increasing hydrostatic pressure (P ) an initially flat square sheet is folded into an approximately spherical ball of diameter R with the packing density 6 hL? where h and L are the sheet thickness and edge size, respectively. As the compaction ratio K -L / R in creases, the packing density increases up to p = 1 at = ( n L / 6 h ) l/3.…”