Experimenting with Elastic Knots

Even though most of us tie our shoelaces wrongly (i.e., not with the most mechanically performant configuration), knots in ropes and filaments have been used for millennia as functional structural mechanisms. Applications include mooring ships to docks, ensuring the safety of a falling climber, wrapping presents or fastening surgical suture threads. Despite their importance, our knowledge on tight physical knots is mostly empirical and there is a striking lack of predictive models that quantitatively capture their mechanical behaviour. For tight knots, highly nonlinear and coupled behaviour arise from intricate three-dimensional geometries, large deformations at the material level, (self)contact and friction between rod strands. Furthermore, tight knots do not exhibit separation of the relevant length scales, thereby precluding usage of thin rod models (e.g., ‘à la’ Kirchhoff). We have approached the problem by performing precision model experiments that comprise X-ray computed tomography (to probe the underlying complex geometry) and mechanical testing (to quantify the interplay of bending curvature, elasticity, and friction). These experimental data are contrasted to both Finite Element (FE) simulations and to formal results for geometrically rigid strings (that exclude elasticity), finding that the latter tends to act as an underlying ‘backbone’ for the full elastic solution. Through the combination of X-ray tomography and FE modeling, we gained quantitative access to the regions of self-contact where friction is activated to confer functional performance, e.g., holding strength. Our results are starting to yield invaluable insight that we hope will help unravel the highly nonlinear but fascinating behaviour of knotted structures.

Work done in collaboration with Paul Grandgeorge (EPFL), Paul Johanns (EPFL), Tomohiko Sano (EPFL), Changyeob Baek (MIT/EPFL), Harmeet Sing (EPFL), and John Maddocks (EPFL).