91福利

In his celebrated 1638 Dialogues Concerning Two New Sciences, Galileo identified a fascinating problem in the mechanics of ropes. His fictitious discussant Salviati asks "How are fibres, each not more than two or three cubits in length, so tightly bound together in the case of a rope one hundred cubits long that great force is required to break it?" He then proceeds to explain that " [...] in the case of the rope the very act of twisting causes the threads to bind one another in such a way that when the rope is stretched with a great force the fibres break rather than separate from each other." With the benefit of hindsight, one might say Galileo recognized that the mechanical integrity of ropes (and by implication staple yarns and woven fabrics) is down to frictional contacts between fibres. But beyond this general observation, and despite our everyday familiarity with these issues, one can argue that Galileo's physics problem has remained unresolved for nearly four hundred years. Here it is proposed that the mechanical integrity of such fibre assemblies is actually a consequence of a generic tensile stress percolation transition, which appears under the Amontons-Coulomb friction laws for long enough fibres and with enough entanglement. This is demonstrated in abstract yarn models in which the friction laws are formulated as a linear programming (LP) problem. In these models the percolation transition is manifest as the onset of LP feasibility, wherein the yarn can in principle support an unbounded tensile load without slippage even though the fibre ends remain tension-free.