Ivars Peterson's MathTrek
May 26, 2003
Crumpling is a ubiquitous, though poorly understood, physical phenomenon. It occurs when a fender absorbs the energy of a car crash, when Earth's crust buckles at the interface between colliding tectonic plates to create a mountain range, when a blood cell's membrane folds to allow the cell to pass through a narrow capillary, when a grape shrivels to a raisin as it dries out, and when a storage tank collapses.
It takes energy to crumple a sheetto force it uniformly into a smaller and smaller ball. Where does that energy go?
Crumpling a sheet of paper into a tight ball causes some regions to become strongly creased, creating a network of narrow ridges and sharp, conical peaks. The material at these creases bends and stretches in response to the applied force.
In the ridges
Several years ago, to investigate the energy involved in crumpling, physicists Thomas A. Witten and Alexander E. Lobkovsky of the University of Chicago and their coworkers developed a computer model to determine what happens at a single ridge between two sharp points.
The researchers discovered that the energy that goes into bending and stretching a material as it is crumpled is concentrated in the network of narrow ridges rather than being stored in the points or distributed more broadly throughout the sheet.
This finding suggests that crumpled sheets can be described in terms of the pattern of ridges and peaks that cover the surface. By adding together the deformation energies associated with individual ridges, scientists can estimate the total energy stored in a given crumpled sheet.
Because sheets of different materialswhether paper, Mylar, or metaldisplay similar ridges, the basic crumpling process must be similar in these materials. The only condition is that the sheets be large and thin.
The researchers also discovered that increasing a sheet's size has an unexpectedly small effect on the total amount of energy required to crumple it. For instance, it takes only twice as much energy to crumple a sheet whose sides are eight times longer, even though the sheet's area is 64 times larger and the ridges are eight times longer.
To get a somewhat different perspective on crumpling, James P. Sethna, Paul A. Houle, and their collaborators at Cornell University focused on the collective behavior of a large number of creases.
These studies were originally motivated by the finding that widely varying physical systems emit pulses of energy comparable to the audible pops and clicks generated by crumpling paper or crinkling cellophane. For example, earthquakes generate irregular trains of seismic shocks, and sudden changes in the magnetic fields trapped in various materials produce bursts of electromagnetic radiation.
Paper crumpling offers a convenient, economical means of studying these effects in the laboratory. By recording and analyzing the sounds generated during crumpling, researchers can estimate the associated energy emissions.
In their experiments, Houle and his coworkers measured the energy of each pop associated with crumpling a sheet of paper and plotted the number of pulses detected at different energies.
Surprisingly, the distribution of pulse energies stayed roughly the same throughout the crumpling process, even though the sheet started out completely flat and ended up wildly wrinkled.
Moreover, different types of paper and different sheet sizes generated remarkably similar energy distributions. This finding applied equally to freshly crumpled sheets, to previously crumpled sheets that were crumpled again, and to sheets that had been folded into a regular, gridlike pattern of creases and smoothed out before crumpling.
Other research on the sounds of crumpling provided further insights. Working with Witten and Lobkovsky, Eric M. Kramer discovered that clicks associated with crumpling Mylar sheets came not from the creation of ridges or peaks but from the sudden shift of a ridge-bounded facet from a slightly concave to a slightly convex profile, or vice versa. The work also supported the notion that a large part of the total elastic energy of a crumpled sheet is contained in a network of narrow, stretching ridges.
In 2002, University of Chicago researchers reported an additional crumpling effect. They started with the observation that even the flimsiest of sheets, when crumpled into a ball, can be extraordinarily resistant to pressure. For example, no matter how hard you squeeze on a crumpled sheet of standard notebook paper, the resulting ball is still about 75 percent air.
In pondering what gives the resulting crumpled ball its surprising strength, the researchers experimented with Mylar. They tested the notion that extra squeezing would require increasing the number of ridges that need to be broken to make a crumpled ball smaller.
To their surprise, they discovered that, when squeezed by a constant force, such a ball doesn't quickly collapse to a final fixed size. Instead, it continues to shrink for periods of up to 3 weeks. Somehow, the crumpled ball dissipates energy over time to allow it to collapse further.
In an award-winning mathematics project at this year's Intel International Science and Engineering Fair, a team of three students focused on ridge lengths in crumpled sheets. Andrew Leifer, David Pothier, and Raymond To of Fairview High School in Boulder, Colo., analyzed the distribution of the lengths of ridges.
The students observed that when a single ridge buckles into smaller ridges, the phenomenon displays fractal characteristics. In effect, a magnified view of a small region of a crumpled sheet shows roughly the same pattern of buckled ridges seen on a larger scale.
Such as pattern can be described in terms of mathematical relationships known as power laws, where one variable is expressed in terms of another variable raised to some power. In previous work, for example, researchers had found that a power law describes the relation between the height of a crumpled ball and the mass of the piston compressing it or the relation between the energy required to buckle a paper ridge and the ridge's length.
The student team sought a mathematical model to describe the frequency distribution of ridge lengths in crumpled paper. Using nothing more than paper, scissors, a ruler, a magnifying glass, a flashlight, and a pneumatic buckler made of LEGO blocks, the students measured and recorded ridge lengths found in various crumpled sheets.
In analyzing their data, they discovered three new power-law relationships in the properties of a crumpled ball: between the number of ridges and the radius of the crumpled ball, between the mean ridge length and the radius, and between the mean ridge length and the number of ridges.
"These power-law results further affirm the fractal nature of the crumpling process," the students concluded in their report. The buckling of a single ridge produces new ridges with ridge lengths that follow a power-law distribution.
Moreover, they found that the frequency distribution of ridge lengths in crumpled paper fits the so-called Weibull distribution. The Weibull distribution resembles an asymmetric bell curve. Their result supported the notion that paper crumpling can be viewed as a repetitive process of buckling multiple ridges and their daughter products.
"Our work has many applications," Leifer, To, and Pothier pointed out in their report. "The distribution of ridge lengths could be combined with previously known theoretical equations to approximate energy absorption." That could have value in the study and design of energy-absorbing materials for packaging, vehicle fenders, and more.
Copyright 2003 by Ivars Peterson
Cerda, E., et al. 1999. Conical dislocations in crumpling. Nature 401(Sept. 2):46-49. Abstract available at http://dx.doi.org/10.1038/43395.
Gompper, G. 1997. Patterns of stress in crumpled sheets. Nature 386(April 3):439-441.
Houle, P.A., and J.P. Sethna. 1996. Acoustic emission from crumpling paper. Physical Review E 54(July):278-283. Abstract available at http://link.aps.org/abstract/PRE/v54/p278. Preprint available at http://arxiv.org/abs/cond-mat/9512055.
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Kramer, E.M., and A.E. Lobkovsky. 1996. Universal power law in the noise from a crumpled sheet. Physical Review E 53(February):1465-1469. Abstract available at http://link.aps.org/abstract/PRE/v53/p1465.
Leifer, A.M., R.C.-H. To, and D.G. Pothier. 2003. Fractals, power-laws and the Weibull distribution: Mathematically modeling crumpled paper. Intel International Science and Engineering Fair. May 11-17. Cleveland.
Lobkovsky, A., and T.A. Witten. 1997. Properties of ridges in elastic membranes. Physical Review E 55(February):1577-1589. Abstract available at http://link.aps.org/abstract/PRE/v55/p1577.
Lobkovsky, A., et al. 1995. Scaling properties of stretching ridges in a crumpled elastic sheet. Science 270(Dec. 1):1482-1485. Abstract available at http://www.sciencemag.org/cgi/content/abstract/270/5241/1482.
Matan, K., et al. 2002. Crumpling a thin sheet. Physical Review Letters 88(Feb. 18):076101. Abstract available at http://link.aps.org/abstract/PRL/v88/e076101. Preprint available at http://xxx.lanl.gov/abs/cond-mat/0111095.
Peterson, I. 1996. The sounds of crumpling. Science News 149(June 15):376-377.
Venkataramani, S. 1998. Cones, creases and crumpled sheets. Physics World (July):19-20.
Tarnai, T. 1997. Folding of uniform plane tessellations. In Origami Science & Art: Proceedings of the Second International Meeting of Origami Science and Scientific Origami, K. Miura, ed. Otsu, Japan: Seian University of Art and Design.
Information about the Intel International Science and Engineering Fair is available at http://www.sciserv.org/isef/.
Information about Paul Houle's research on crumpling can be found at http://www.honeylocust.com/crumpling/ and http://simscience.org/crackling/Intermediate/Paper/PaperCrackles.html.
Information about research at the University of Chicago on forced crumpling is available at http://jfi.uchicago.edu/~tten/rainbow/Crumpling/.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles. Find it at the MAA Bookstore.