Uniqueness/existence for FEA with Amontons friction?

[bcrowell]

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...?

 

[eljose79]

Dear forum post,

Thank you for sharing your research project involving numerical simulations of knots. It sounds like a fascinating and challenging project.

In response to your question about the existence and uniqueness of motion when using the Amontons model of friction, it is important to consider the assumptions and limitations of this model. The Amontons model is a simple and widely used model of friction, but it does have its limitations. One of the main limitations is that it assumes a constant coefficient of friction, which may not be accurate for all types of materials and surfaces.

In terms of your specific system of a rope with multiple segments in contact, it is possible that the Amontons model may not give a uniquely defined answer. In this case, it may be necessary to consider other factors, such as the surface roughness of the rope and the materials involved, to accurately simulate the frictional forces at play.

Additionally, as you mentioned, there are various software packages available for FEA that may have different methods for handling this issue. It may be worth exploring different software options and consulting with experts in the field to determine the best approach for your specific research project.

Overall, it is important to carefully consider the assumptions and limitations of the Amontons model of friction in your simulations and to explore other factors that may affect the frictional forces in your system. I wish you the best of luck in your research project.