A crumpled sheet of paper shows two recurrent structures: cones and ridges. These are the pivotal hinges that allow the compression of the surface into a smaller volume. As the volume is decreased, the surface bends around point singularities or cones and those points are connected through lines of high curvature or ridges.

We have shown that cones can be mathematically described by using the famous Elastica equation. Our approach explains different situations for the early stages of crumpling where a network of ridges has not been developed yet. Some examples we are looking at are the confinment of a sheet into a cylindrical frame, the drape of a surface hanged from one fixed point, and the axysimmetric rotation of a thin plate.

People involved

Eugenio Hamm (Universidad de Santiago)
Tao Liang (University of Chicago)
Marcel López (Universidad de Santiago)
L. Mahadevan (Harvard University)
Jordan Weil (University of Chicago)
Thomas Witten (University of Chicago)
Enrique Cerda (Universidad de Santiago)

References

"Asymptotic Shape of a Fullerene Ball", Witten, T. and Li, H. 1993. EuroPhys. Lett., 23, 51.

"Conical dislocations in crumpling", Cerda, E. , Chaieb, S., Melo, F. and Mahadevan, L. 1999. Nature, 401, 46 - 50.

"The elements of draping", Cerda, L. Mahadevan and J.M. Passini, 2004. PNAS, 101, 1806.

"Confined developable elastic surfaces: cylinders, cones and the Elastica", Cerda, E., and L. Mahadevan, 2005. Proc. R. Soc. A, 461, 671.

"Spontaneous curvature cancellation in forced thin sheets", Liang T. and Witten T. 2006. Submitted to Phys. Rev. E.