A model for the fragmentation kinetics of crumpled thin sheets

Abstract

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.

Simulation of crumpled sheets via alternating quasistatic and dynamic representations

“…Through repeated, uniaxial crumpling of Mylar sheets in a cylindrical setup, the authors found that the accumulated crease length evolves logarithmically in the number of crumpling cycles. In a subsequent study, a physical explanation for this robust logarithmic scaling was proposed by relating crumpling to a fragmentation process that dictates the subdivision of a sheet's surface into smaller facets over time [27]. The statistics of facet area in each crease pattern were consistent with an area-conserving fragmentation rate equation used previously to describe fracture phenomena [28].…”

“…The radially confined sheet has an effective thickness of 0.125 mm, while the remaining examples are 0.25 mm. Experimental studies of repeated cylindrical confinement have revealed remarkably robust properties in the accumulation of total crease length and the distribution of facet sizes, as previously described [26,27]. A possible application of numerical simulation would be to verify whether these statistical properties are universal to different modes of crumpling such as radial confinement, which may otherwise pose experimental challenges.…”

Thermally fluctuating, semiflexible sheets in simple shear flow

“…23 The dynamics of crumpling 2D sheets (both athermal and thermal) in different contexts has also received attention in recent years. 77,78 In order to study the conformational behavior of sheets as a function of S and (k B T/k), we consider two geometric quantities: minimumvolume bounding ellipsoids and orientational covariance matrices. Minimum-volume bounding ellipsoids of sheets were calculated for snapshots of sheet configurations every _ gDt snap = 0.25 units of time via the following (dual) convex optimization problem: 79…”

Surface Tension Scars Soft Solids

“…The discovery of an alternative mechanism for surface scarring in a similar (though not mechanically identical) way to how paper creases when repeatedly crumpled [6] could also have practical applications-for example, in soft-robotics devices that fold [7] or in surface engineering for fluid transport [8]. Indeed, one could envision manipulating the surface tension of soft solids to "program" their creasing pattern with a view to controlling the transport of liquids over such surfaces.…”