How Do Bumps Form in Carpets?

We’ve all had this experience: we are walking on a carpet, and we suddenly trip over an annoying bump (or “ruck”) that we didn’t know was there. So how did it form?

My colleagues Alpha Lee, Clément Le Goullec, and Dominic Vella from the Mathematical Institute at University of Oxford have just posted a new paper that endeavors to explain an apparent paradox in the formation of carper rucks.

As the authors write in their abstract:

Everyday experience suggests that a ‘ruck’ forms when the two ends of a heavy carpet or rug are brought closer together. Classical analysis, however, shows that the horizontal compressive force needed to create such a ruck should be infinite. We show that this apparent paradox is due to the assumption of inextensibility of the rug. By accounting for a finite extensibility, we show that rucks appear with a finite, non-zero end-shortening and confirm our theoretical results with simple experiments. Finally, we note that the appropriate measure of extensibility, the stretchability, is in this case not determined purely by geometry, but incorporates the mechanics of the sheet.

Figure 1 from the paper by Lee et al.