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I'm exploring a possible research project involving numerical simulations of knots. For example, I would like to be able to determine from simulations what is the lowest coefficient of friction for which a square knot holds under load. (Pencil-and-paper calculations by Maddocks and Keller with some approximations give about 0.24, but the approximations are probably not very good.) The idea is to do a finite element analysis (FEA), using the Amontons model of friction (i.e., the model of friction taught in freshman physics).

I'm now trying to sketch out what a physical model would look like, and it's not really obvious to me that the Amontons model gives a uniquely defined answer for this type of system. My idea was to do a very simple FEA in which I break up the rope into short segments (e.g., a straight rope would be a series of disks). A given segment of the rope could be in contact with as many as six other segments. We tell freshman physics students that static friction acts in the direction that tends to prevent slipping, but this seems likely to leave me with an underdetermined system, since we have as many as six frictional forces, each with two unknown components in the plane of contact, for a total of as many as 12 degrees of freedom.

Is there any reason to expect existence and uniqueness of the motion when Amontons friction is used? There are various software packages to do general-purpose FEA, so it seems like there must be some way of handling this issue...?

I'm now trying to sketch out what a physical model would look like, and it's not really obvious to me that the Amontons model gives a uniquely defined answer for this type of system. My idea was to do a very simple FEA in which I break up the rope into short segments (e.g., a straight rope would be a series of disks). A given segment of the rope could be in contact with as many as six other segments. We tell freshman physics students that static friction acts in the direction that tends to prevent slipping, but this seems likely to leave me with an underdetermined system, since we have as many as six frictional forces, each with two unknown components in the plane of contact, for a total of as many as 12 degrees of freedom.

Is there any reason to expect existence and uniqueness of the motion when Amontons friction is used? There are various software packages to do general-purpose FEA, so it seems like there must be some way of handling this issue...?

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