Mathematicians and mechanical engineers from EPFL have collaborated to gain better insights into the mechanics and geometry of two filaments in contact—as is the case of woven fabrics and knots.
Pedro Reis, head of EPFL’s Flexible Structures Laboratory, and John Maddocks, head of EPFL’s Laboratory for Computation and Visualization in Mathematics and Mechanics, have a common interest—a fascination with ropes and knots.
Reis is an engineer and an avid rock climber, while Maddocks is a mathematician and has a passion for sailing. However, their mutual interest in knots is not limited to their hobbies. This is because knots find use in a range of applications, for instance, surgical sutures. Knots have been part of the daily lives of humans since time immemorial, but their mechanics are yet understood completely.
A Simplified Knot
Reis, Maddocks, and the researchers in their labs have investigated a particular configuration of contact between two filaments, or the orthogonal clasp, which can be considered as the most fundamental building block for any knot.
This intertwining is the simplest of all knots; or to be more specific, it’s the link that knots are based on. It’s also the most widely used knot. It’s found in the thread patterns in our clothing, for example.
Pedro Reis, Head, Flexible Structures Laboratory, EPFL
Reis, Maddocks, and their group performed a comprehensive analysis of the contact region between the two filaments. The study results were published recently in Proceedings of the National Academy of Sciences of the United States of America (PNAS).
Maddocks has been analyzing (besides other things) the mathematical theories explaining the mechanics of knots for more than three decades. He has been studying specifically the complex geometry of the curves constituting the contact region between filaments. Eugene Starostin, Maddocks’ colleague at the time, published a paper in 2003, which deals particularly with the orthogonal clasp.
In this kind of clasp, the contact zone resembles a diamond shape, and the four corners denote the main pressure peaks. But his theory could never be empirically verified because of technical limitations.
When Pedro and I decided to work together, we wondered whether Starostin’s earlier results would still be relevant in practice. We then carried out tests, measurements and experiments to answer this question.
John Maddocks, Head, Laboratory for Computation and Visualization in Mathematics and Mechanics, EPFL
“The contact region has always been calculated according to an ideal hypothesis, but never experimentally verified,” added Reis.
Verifying the Initial Findings
At Reis’ lab, researchers performed experiments with the help of a tomograph, which uses computer models and X-rays to produce 3D images of objects.
“Tomography lets us look inside the contact region between the two filaments. We then corroborated our experimental results with computer simulations. We didn’t expect to find such a heterogeneous pressure distribution between the two filaments,” noted Paul Grandgeorge, a postdoc at Reis’ lab.
The experiments demonstrated that the pressure region between both filaments coincided with Starostin’s previous geometrical calculations. “This is a small step forward in understanding filaments in contact,” added Maddocks.
The Capstan Equation
The researchers were motivated by the findings and intended to go a step ahead. Thus, they analyzed the contact region between two filaments under the effect of friction. Their initial speculation was that friction could be described by using the capstan equation.
The concept behind the capstan equation is simple: when a rope is wrapped around a cylindrical tube, such as a mooring bollard, the tensions in the two hanging strands are separated. The more loops there are around the tube, the greater the difference in tension between the two strands. We assumed that we could use this equation to calculate the tension ratio between the two strands in our experiments.” says Grandgeorge.
Paul Grandgeorge, Postdoc, Flexible Structures Laboratory, EPFL
But after performing a number of experiments, the researchers summarized that the capstan equation cannot help in the case of filaments in a frictional state.
“The capstan equation assumes that the tube does not deform and is larger in diameter than the rope wrapped around it. In our experiments, however, the elastic rod that functions as the tube can be deformed and has the same diameter as the second rod that functions as the rope,” added Reis.
However, the researchers do not consider these results as a setback, but exactly the opposite. “This gives us even more incentive to find a theoretical model that can explain this physical phenomenon,” noted Grandgeorge. “It may appear to be a simple problem, but geometrically it’s actually quite complicated,” explained Maddocks.
The researchers predict that there will be numerous studies on knots in the future, which will offer better theoretical insights into real-world situations.
Theory and experiments to understand the contact between two filaments
Video Credit: EPFL.
Grandgeorge, P., et al. (2021) Mechanics of two filaments in tight orthogonal contact. Proceedings of the National Academy of Sciences of the United States of America. doi.org/10.1073/pnas.2021684118.