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a crumpled 2-foot-high mountain made of 1/2 mil mylar 32 k bytes The stretching ridges in the sheet supply enough rigidity for the for 30 grams of material to span a volume over 2000 times that of the mylar, with no supporting structures. |
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Tetrahedra that pack to fill space. Download us. Print copies. Cut us out. Tape us together. Fill space. pdf file |
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Spontaneous free-boundary structure in crumpled membranes: We investigate the strong curvature that appears at the boundaries of a thin crumpled elastic membrane. We account for these high-curvature regions in terms of the stretching-ridge singularity believed to dominate the structure of strongly deformed elastic membranes. Using a membrane fastened to itself to form a bag shape with a single stretching ridge, we show that the creation of points of high boundary curvature lowers the interior ridge's energy. In the limit of small thickness, the induced curvature becomes arbitrarily strong on the scale of the object size and results in induced ridges connecting interior vertices to the boundary. We find that small-scale truncation near the vertex of a ridge can influence its energy greatly. preprint of invited paper for DeGennes memorial issue in J. Phys. Chem. http://arxiv.org/abs/0808.3759 |
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 | Force focusing in confined fibers and sheets: A sheet of office paper coiled into a mailing tube hugs the wall of the tube in order to minimize its bending. But the contact with the wall is incomplete; near the edge, the paper detaches or takes off from the wall and rejoins the cylinder only at the edge. Such detachment is a commonplace feature of coiled sheets or fibers small and large. Here we show that the detached region has a universal shape that touches down at an angle of 24.1 degrees. Moreover, the takeoff point experiences a focused force controlled by the length of the fiber or sheet. preprint on cond-mat .4 megabyte. J. Phys. D: Appl. Phys. 41 (2008) 132003, 10.1088/0022-3727/41/13/132003. Nature story. | |
 | Stress Focusing in Elastic Sheets, T. A. Witten. In which many aspects of crumpling singularities are reviewed. preprint pdf, 3.1 megabyte; Reviews of Modern Physics 79 643 (2007), DOI: 10.1103/RevModPhys.79.643 | |
 | Numerical Investigation of Isolated Crescent Singularity, Tao Liang. In which inner a new intermiate length scale is exhibited for a crescent singularity resembling a d-cone. The both crescent central curvature and the crescent transverse curvature are found to scale differently with thickness. The width scaling is constent with that of the Podgorelev ring ridge. http://arxiv.org/abs/cond-mat/0610781 Submitted to Phys. Rev. E October 2006 | |
 | Spontaneous curvature cancellation in forced thin sheets, Tao Liang, Thomas A. Witten. In which the mean curvature at the supporting rim of a d-cone is shown to vanish under a wide range of conditions, via numerical and experimental measurements. http://arxiv.org/abs/cond-mat/0512162, Phys. Rev. E 73 046604 (2006) | |
 | Crescent Singularities in Crumpled Sheets, Tao Liang and Thomas A. Witten. In which the the scaling of the anomalously wide crescent region is investigated. http://arxiv.org/abs/cond-mat/0407466. Phys. Rev. E 71, 016612 (2005) | |
 | Crumpling a Thin Sheet Kittiwit Matan, Rachel Williams, Thomas A. Witten, Sidney R. Nagel Comments: revtex 4 pages, 6 eps figures Phys Rev. Letters 88, 076101 (2002) http://arxiv.org/abs/cond-mat/0111095. Squeezing a crumpled sheet of mylar into a cylinder reveals a surprizing logarithmic relaxation process and a force-vs-compression power law. | |
 | Scaling of the buckling transition of ridges in thin sheets, Brian DiDonna http://arxiv.org/abs/cond-mat/0108312 submitted to Physical Review E 66, 016601 (2002). Conventional buckling plate analysis leads to numerically confirmed predictions about when, where, how and why a ridge buckles. | |
 | Trapping of Vibrational Energy in Crumpled Sheets Ajay Gopinathan, T.A. Witten, S.C. Venkataramani http://arxiv.org/abs/cond-mat/0109059 Physical Review E. 65 036613 (2002). Elastic wave analysis and simulations show that vibrational energy should get trapped in the faces of crumpled sheets. | |
 | Anomalous strength of membranes with elastic ridges B. A. DiDonna and T. A. Witten, ...in which we show that the buckling strength of ridges is controlled by the same scaling laws that govern its resting energy, at Physical Review Letters, 87 206105 (2001). Also at http://arxiv.org/abs/cond-mat/0104119 11/10/01 | |
 | Singularities, structures, and scaling in deformed m-dimensional elastic manifolds, B. A. DiDonna, S. Venkataramani, T. A. Witten and E. M. Kramer, ...in which we demonstrate two distinct forms of energy condensation depending on the embedding dimension, at http://xxx.lanl.gov/abs/math-ph/0101002, Physical Review E 65, 016603 (2002) | |
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Limitations on the smooth confinement of an unstretchable manifold, a math paper showing that an M dimensional sheet can't fit into a small sphere without stretching or folding in a world of fewer than 2M dimensions | |
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Stress condensation in crushed elastic manifolds, Eric M. Kramer and
Thomas A. Witten
Phys. Rev. Lett.
78 1303-1306 (1997).
LANL Archive abstract |
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Alex Lobkovsky: "Structure of crumpled thin elastic membranes, PhD Dissertation, University of Chicago, August, 1996 gzipped postscript, 400 K, Adobe pdf, 1600 K |
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"Properties of Ridges in Elastic Membranes" Alexander E. Lobkovsky and T. A. Witten, Physical Review E 55 1577-1589 (1997) eprint archive: cond-mat/9609068 |
When a thin elastic sheet is confined to a region much smaller than
its size the morphology of the resulting crumpled membrane is a network
of straight ridges or folds that meet at sharp vertices. A virial theorem
predicts the ratio of the total bending and stretching energies of a ridge.
Small strains and curvatures persist far away from the ridge. We discuss
several kinds of perturbations that distinguish a ridge in a crumpled sheet
from an isolated ridge studied earlier (A.~E. Lobkovsky, Phys. Rev. E.
{\bf 53} 3750 (1996)). Linear response as well as buckling properties are
investigated. We find that quite generally, the energy of a ridge can change
by no more than a finite fraction before it buckles.
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Universal Power Law in the Noise from a Crumpled Elastic Sheet. Eric M. Kramer and Alexander E. Lobkovsky Phys Rev E. . 53 1465 (1995)PDF |
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Scaling properties of stretching ridges in a crumpled elastic sheet |
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"Asymptotic Shape of a Fullerene Ball,"Europhys. Lett 23 51-55 (1993) pdf 315 k |
LEAD PARAGRAPH - Crumple a piece of paper, squeezing it into a crooked sphere.
Even the strongest of hands is not able to squeeze it much smaller than a golf ball. A sheet of paper, flimsy when flat, gains surprising strength as it crumples.
This page was accessed about 4 times a day from outside the U of C.in April, 1997
* This material is partially based upon work supported by the National Science Foundation under Grant Nos. DMR 9528597 and DMR 9975533. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
T. Witten, t-witten@uchicago.edu 11/01