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Paper Chase

September 7, 2010

The computer was supposed to be the herald of the age of the paperless office, but that age is very slow in coming. Computers actually increased the consumption of paper. Even today, businesses archive paper copies of 62% of their important documents, since they think they're needed for legal reasons.[1,2] The advantages of electronic versions of documents include easy search, easy storage, and the ability to instantly transmit information throughout the world. I think that electronic book readers and devices like the iPad will finally lead to a mostly paperless lifestyle; provided, however, the vendors don't let digital rights management and incompatible formats spoil the game.

In the context of not seeing the forest for the trees, we all notice the words on a piece of paper, but we don't usually notice the sheet of paper itself, unless it's ostentatious, like a wedding invitation. Aside from the many centuries of technology that's a part of papermaking, there's a lot of physics in a sheet of paper. One of the easiest things you can do to a sheet of paper is to crumple it. Starting a little more than a decade ago, physicists started to investigate crumpling, and they found some universal laws.

One easy experiment is to analyze the sound emitted during crumpling. Of course, you need to be consistent about how you crumple a sheet, and it's that type of problem that keeps experimentalists awake at night. In the mid-1990s, Eric M. Kramer and Alexander E. Lobkovsky of the James Franck Institute and the Department of Physics, The University of Chicago crumpled Mylar sheets and recorded the emitted sound, which appeared as discrete clicks.[3-4] It was possible to convert sound amplitude to energy, and the energy involved in the crumpling deformation ranged over six orders of magnitude. The energies of the clicks were distributed according to a power law distribution.
P(E) ∼ e
where P(E) is the probability of a click having energy E, and α is about one. The power law didn't change when the sheet material or size of the sheet was changed, so it's a universal law. Other researchers have found a power law relationship between the height of a crumpled ball and the weight of a compressing mass; between the number of crumple creases and the radius of the ball; between the mean length of a crease and the ball radius; and between the mean length of a crease and the number of creases.[3]

Along with sound, another way to analyze crumpled sheets is by examination of the creases that are formed during crumpling. Lobkovsky, along with another associate, Thomas A. Witten, did a computer model of crumpling, and they found that all the deformation energy is concentrated narrowly in the creases that are formed.[3,5] Since different materials display the same ridge patterns when crumpled, they concluded that the crumpling process is indeed universal. Interestingly, the size of a sheet has only a small effect on the ridge pattern and the energy involved in crumpling. It takes only twice as much energy to crumple a sheet that's eight times larger on a side than another sheet. That's an area difference of sixty four, or nearly two orders of magnitude, with little affect on the crumpling energy. If I were designing crumple zones into an automobile, should I use a single sheet, or should I use multiple sheets that are offset from each other by a small distance? You can see why basic physics is relevant to many industries.

Laser topograph of crumpled paper

Laser topograph of crumpled paper; Daniel L. Blair and Arshad Kudrolli (via arXiv).

Another effect, well known to everyone who's been hit in the head with a wad of paper, is that crumpled balls of paper are mechanically stiff; that is, it takes quite a bit of force to crumple them further, so they bounce off your head, rather than squash. That's remarkable, since a typical crumpled ball of paper is 75% air.[3] Another remarkable feature of crumpled paper balls is that they creep. If you apply a constant force, a crumpled ball will continue to shrink for several weeks.[3]

Instead of crumpling sheets of paper, what if we stretched them, instead? A paper in the September 3, 2010, issue of Physical Review Letters [6-7] reports on experiments in which sheets of paper are clamped at a top edge, loaded with weights at a bottom edge, and left hanging until they tear. A paper engineer would probably measure the distribution of tear time as a function of load. The physicists doing these experiments looked instead at mechanical happenings on a microscopic level. A sheet of paper, of course, is not uniform in thickness at a microscopic level over its area, so you would expect to see differences in mechanical behavior from point to point. What was observed for the paper sheets overall was an initial rapid creep rate, followed by a slower creep rate - What a materials scientist would call strain hardening. What was interesting, however, was the fact that the variation in creep from place to place at the microscopic level increased as time went on. Some places were bearing their load well, while others didn't. Sounds like a model for human organizations, as well.

The Paper Chase (1973, James Bridges, Director) was a movie about a law school student that I was never interested in seeing. It became a television series (1978-1986, Directors, Various, 1978-1986), perhaps notable for an occasional appearance by a young Jane Kaczmarek and a young Jon Lovitz. I didn't watch any of that, either. The only law show I ever enjoyed was Perry Mason, but I'm showing my age.


  1. Thom Patterson, "Is the 'paperless' office here at last?" CNN, March 18, 2010.
  2. Abigail J. Sellen and Richard H. R. Harper, "The Myth of the Paperless Office," MIT Press (April 1, 2003), pp. 245.
  3. Ivars Peterson, "Deciphering the Wrinkles of Crumpled Sheets," Ivars Peterson's MathTrek (Mathematical Association of America, May 26, 2003).
  4. Eric M. Kramer and Alexander E. Lobkovsky, "Universal power law in the noise from a crumpled elastic sheet." Phys. Rev. E, vol. 53 (1996), pp. 1465-1469.
  5. Alexander E. Lobkovsky and T. A. Witten, "Properties of ridges in elastic membranes," Phys. Rev. E, vol. 55 (1997), pp. 1577-1589.
  6. Jari Rosti, Juha Koivisto, Lasse Laurson and Mikko J. Alava, "Fluctuations and scaling in creep deformation," arXiv (July 27, 2010).
  7. David Lindley, "Paper Stretches Unevenly," Physical Review Focus, August 27, 2010.
  8. Michael Marder, Robert D. Deegan and Eran Sharon, "Crumpling, buckling, and cracking: Elasticity of thin sheets," Physics Today, vol. 60, no. 2 (February 2007), pp. 33-38.

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Linked Keywords: paperless office; paper; electronic book; iPad; digital rights management; not seeing the forest for the trees; papermaking; physics; experimentalists; James Franck Institute; Department of Physics, University of Chicago; mylar; power law distribution; deformation; crumple zones; Laser topograph of crumpled paper; mechanically stiff; creep; Physical Review Letters; strain hardening; The Paper Chase; Jane Kaczmarek; Jon Lovitz; Perry Mason.