Forced Crumpling* at University of Chicago

	shapiness: demonstration of the shape memory of a crumpled sheet.
	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.
	Tetrahedra that pack to fill space. Download us. Print copies. Cut us out. Tape us together. Fill space. pdf file

Eric Kramer's simulated crumpled sheet 

Closing a bag: an induced ridge

The Swallowtail Fold

This detail from Leonardo DaVenci's famous painting of the Mona Lisa shows a characteristic folding pattern seen everywhere in fabric and other thin sheets. Two circles mark the repeating feature of this pattern. It could be called a swallowtail fold. A point of cloth extends outward, flanked by three folds. Two of these form a Λ shape with fabric tucked between them. The third extends upward and back. Our ears have such a fold, with the two tails pointing upward and forward just under the top of the ear. Maybe it functions to rigidify the ear shape.
Thanks for the name Swallowtail fold to Eugenio Hamm.

crumpled chair Naomi found on the internet

if this is your picture, let me know so I can credit you!

The Mowitz Crease animation

A small crease allows buckling into a cone with no stretching

Aaron Mowitz 2020



[if this gif animation doesn't play, try opening it in a new tab]

The crease is an arc of a circle. The opening angle profile and the torsion of the crease are specified in a parameterized form that guarantees a smooth surface around a terminating crease. The rest of the sheet deforms without stretching, with the straight generator lines shown. The parameters in the opening angle and torsion functions are adjusted to minimize the bending energy of the surface. The two colors denote the sheet spreading from the two sides of the crease. The straight lines are the generators of the sheet. The surface is smooth at the junction of the two sides and the curvature is continuous. see paper below

if animation doesn't play, try opening it in a new tab

Our work

	Finite curved creases in infinite isometric sheets, Aaron J. Mowitz      see animation above Geometric stress focusing, e.g. in a crumpled sheet, creates point-like vertices that terminate in a characteristic local crescent shape. The observed scaling of the size of this crescent is an open question in the stress focusing of elastic thin sheets. According to experiments and simulations, this size depends on the outer dimension of the sheet, but intuition and rudimentary energy balance indicate it should only depend on the sheet thickness. We address this discrepancy by modeling the observed crescent with a more geometric approach, where we treat the crescent as a curved crease in an isometric sheet. Although curved creases have already been studied extensively, the crescent in a crumpled sheet has its own unique features: the material crescent terminates within the material, and the material extent is indefinitely larger than the extent of the crescent. These features together with the general constraints of isometry lead to constraints linking the surface profile to the crease-line geometry. We construct several examples obeying these constraints, showing finite curved creases are fully realizable. This approach has some particular advantages over previous analyses, as we are able to describe the entire material without having to exclude the region around the sharp crescent. Finally, we deduce testable relations between the crease and the surrounding sheet, and discuss some of the implications of our approach with regards to the scaling of the crescent size. https://arxiv.org/abs/2012.04834.  Physical Review E, accepted February 2022 Aaron's PhD thesis (Local copy) is an extended version of this paper.
	Confined disclinations: exterior vs material constraints in developable thin elastic sheets, Efi Efrati, Luka Pocivavsek, Ruben Meza, Ka Yee C. Lee, Thomas A. Witten      In which the excess gaussian charge of an e-cone is combined with the external conical or planar constraints of a d-cone to create a large class of structures governed by a single combination of the gaussian charge and the cone angle. In the singular limit where the cone angle goes to zero (planar support), buckling away from this support becomes particularly robust and striking.  http://arxiv.org/abs/1410.3531.  Phys. Rev. E 91,(2015).
	Shape and symmetry of a fluid-supported elastic sheet, Haim Diamant and T. A. Witten.  In which a mysterious symmetry of the integrable folding shape of a compressed sheet on a heavy fluid is explored.  We point out the symmetry based on our previous mapping between the folding sheet and a sine-gordan chain.  We identify a generator of this symmetry within the Hamiltonian system that determines the buckling shape.  http://arxiv.org/abs/1304.1937.  Phys. Rev. E 88, 012401 (2013). JFI highlight2014.pdf
	Anomalously fast kinetics of lipid monolayer buckling, Naomi Oppenheimer, Haim Diamant and T. A. Witten.  We re-examine previous observations of folding kinetics of compressed lipid mono-layers in light of the accepted mechanical buckling mechanism recently proposed [L. Pocivavsek et al., Soft Matter, 2008, 4, 2019]. Using simple models, we set conservative limits on a) the energy released in the mechanical buckling process and b) the kinetic energy entailed by the observed folding motion. These limits imply a kinetic energy at least thirty times greater than the energy supplied by the buckling instability. We discuss possible extensions of the accepted picture that might resolve this discrepancy.  Updated on ArXiV 7April 2013.  To be submitted to Phys. Rev. E.
	Compression-induced folding of a sheet: an integrable system, Haim Diamant, Thomas A. Witten.  The apparently intractable shape of a fold in a compressed elastic film lying on a fluid substrate is found to have an exact solution. Such systems buckle at a nonzero wavevector set by the bending stiffness of the film and the weight of the substrate fluid. Our solution describes the entire progression from a weakly displaced sinusoidal buckling to a single large fold that contacts itself. The pressure decrease is exactly quadratic in the lateral displacement. We identify a complex wavevector whose magnitude remains invariant with compression.  http://arxiv.org/abs/1107.5505. pdf graphic 1.9 meg. Phys. Rev. Lett. 107 164302 (2011. Press coverage "Nugget" layman summary
	Rim curvature anomaly in thin conical sheets revisited, Jin Wang.   In which the "spontaneous curvature cancellation"  reported in a previous paper (see below) is found not to hold for sufficiently thin sheets.  For full paper, see http://arxiv.org/abs/1105.1366, Physical Review E., 84 066603, PhD thesis (pdf 10 meg)
	Instability of infinitesimal wrinkles against folding, Haim Diamant, Thomas A. Witten.  In which we show that any compression-induced buckling leading to sinusoidal wrinkling of a sheet on a fluid substrate is unstable against collapse to a finite region of the sheet.  The width of the buckled region scales as the inverse of the compressional displacement, which may be arbitrarily small compared to the buckling wavelength.  For preprint see http://arxiv.org/abs/1009.2487 This paper is superseded by Basile Audoly's paper and by the "integrable system" paper above.
	The compensation of Gaussian curvature in developable cones is local, Jin Wang and T. A. Witten. We report a curious numerical finding that we recently discovered about a d-cone, a thin sheet when pushed into a circular opening. We find that the Gauss Bonnet integral, which must add to zero over the whole sheet, in fact adds to zero within small parts of it. http://arxiv.org/abs/0902.0018 Submitted to Phys. Rev. E January 2009
	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 sharp edges connecting interior vertices to the boundary. REVISED: We analyze these edges as conical sectors with no stretching. As the membrane size diverges, the edge energy grows as the square root of the central ridge energy. For comparison, we discuss the effect of truncating a stretching ridge at its ends. The effect of truncation becomes appreciable when the truncation length is comparable to the width of the untruncated ridge. preprint of invited paper for DeGennes memorial issue in J. Phys. Chem. http://arxiv.org/abs/0808.3759 11/22/08: The process of responding to the reviewers' comments led to major revisions and altered conclusions. The revised version is under copyright by the Journal of Chemical Physics and may not be posted here or on ArXiv. The original ArXiv preprint denoted v1 is still of interest though some of its conclusions are not borne out. These changes have been indicated in the revised abstract above. --T. Witten
	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. Cerda video
	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 See later work
	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) Physical Review E. For resolution of this puzzle, see "Rim curvature anomaly..." above.
	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)
	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
	Stress condensation in crushed elastic manifolds, Eric M. Kramer and Thomas A. Witten Phys. Rev. Lett. 78 1303-1306 (1997). LANL Archive abstract

Alex Lobkovsky: "Structure of crumpled thin elastic membranes, PhD Dissertation, University of Chicago, August, 1996 gzipped postscript, 400 K, Adobe pdf, 1600 K

"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.
 
 

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

 

Scaling properties of stretching ridges in a crumpled elastic sheet

"Asymptotic Shape of a Fullerene Ball,"Europhys. Lett 23 51-55 (1993) pdf 315 k

 

stories on our work

Our people

• Efi Efrati, Weismann Institute
• Eric Kramer
• Alex Lobkovsky
• Tom Witten
• Shankar Venkataramani
• Brian DiDonna
• Ajay Gopinathan
• Tao Liang
• Prof. Sidney Nagel
• Kittiwit Matan

Related developments elsewhere

• Emma McNally is the science photographer who made the picture on the left
• Martin Nettelbladt's Geometry of bending celebrates the shapes of bending, twisting and writhing rods and strips
• Eugenio Hamm professor of physics at Universidad de Santiago de Chile.
• Sahraoui Chaieb at University of Illinois studies scaling of d-cone defects
• Erique Cerda's research page from Chile
  Fiber Coiling Inside a Tube, a you-tube video. As the tube shrinks down the fiber curls further and further around the tube, making an intricate pattern of bounces and contact zones (red).
  
  check for other you-tube videos by ecerdaVI
• Paul Houle at Cornell physics department studies Acoustic emission from crumpling paper
• The States of Tethered Membranes, by Farid Abraham, IBM San Jose
• Fabric draping at the Cornell Theory Center: Bijian Chen and Muthu Govindaraj
• real-world crumpling: from the Ranpak Corporation in Ohio, makers of packaging material.
• Teabag problem from Andreas Gammel. Take 2 square pieces of paper of width 1. Put them exactly on top of each other, and glue the sides together (just the sides, not the surfaces) . This results in a sealed 'teabag'. Make a small hole in it and blow air into the teabag. The result will be a pillow shaped thingie. Question: What is the maximum volume that the teabag can have, if it can bend but not stretch. That is, the Gaussian curvature must be zero everywhere. [TW: does it have to stay flat or can it bulge? Andreas says it can buldge.]
• Richard Turk's test biological material that wrinkles

* This material is partially based upon work supported by the National Science Foundation. 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 Foundation (NSF).

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