# Uniqueness/existence for FEA with Amontons friction?

• bcrowell
In summary, Amontons friction is a constant coefficient of dry friction that is used in FEA to model resistance between two surfaces in contact. Uniqueness in FEA with Amontons friction is determined by continuity and compatibility conditions, and can be affected by factors such as contact stiffness, geometry, and applied loads. Non-uniqueness can be addressed by using a friction model that considers adhesion and shear strength. However, there are limitations to using Amontons friction in FEA, such as its accuracy for small contact areas and low sliding velocities, and its inability to account for wear and temperature effects. More advanced friction models may be necessary in certain cases.
bcrowell
Staff Emeritus
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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...?

Last edited:

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.

## 1. What is Amontons friction and how does it affect FEA?

Amontons friction is a type of dry friction that occurs between two surfaces in contact. It is characterized by a constant coefficient of friction, regardless of the applied load or surface area. In FEA, Amontons friction is modeled using a contact algorithm to account for the resistance between two surfaces in contact.

## 2. How is uniqueness determined in FEA with Amontons friction?

In FEA with Amontons friction, uniqueness is determined by the continuity and compatibility conditions at the contact interface. These conditions ensure that the forces and displacements at the interface are in equilibrium and that there are no gaps or overlaps between the surfaces in contact.

## 3. What factors can affect the existence of a solution in FEA with Amontons friction?

The existence of a solution in FEA with Amontons friction can be affected by several factors, including the contact stiffness, geometry and material properties of the surfaces in contact, and the applied loads and boundary conditions. Additionally, the choice of contact algorithm and mesh refinement can also impact the existence of a solution.

## 4. How can non-uniqueness be addressed in FEA with Amontons friction?

Non-uniqueness in FEA with Amontons friction can be addressed by using a friction model that accounts for the effects of adhesion and shear strength. Additionally, adjusting the contact stiffness and mesh refinement can also help to improve the uniqueness of the solution.

## 5. Are there any limitations to using Amontons friction in FEA?

Yes, there are some limitations to using Amontons friction in FEA. This type of friction is only accurate for small contact areas and low sliding velocities. It also does not account for the effects of wear and temperature on the contact interface. In some cases, more advanced friction models may be needed to accurately capture the behavior of the contact interface.

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