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There are tons of information about the transition from static friction to kinetic friction but I'm having a little trouble finding information about the converse situation.

I need that because I'm simulating an oscillator damped by friction between two surfaces, but I'm in doubt if static friction will even develop properly between two surfaces that nearly don't stop relative to each other.

How do I know if it will be stopped for "long enough" and how the transition from kinetic to static friction happens?

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While there is a transition from static friction to kinetic friction surface I don’t believe there is theoretically any transition from kinetic to static. That’s because kinetic friction is theoretically a constant force that does not depend on the applied force it opposes. It equals $uN$ where $u$ is the coefficient of kinetic friction and $N$ is the force normal to the surface.

If the applied force is removed from a sliding object the net force on the object is the kinetic friction force causing it to decelerate and eventually stop.

Hope this helps.

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All of the models are empirical derived after careful and messy testing. There isn't (yet) a good theoretical basis for friction as far as I know.

The best data we have is from the Stribeck Curve which examines friction as a function of speed for partially lubricated surfaces. Some curves also include non-lubricated contact.

The curve typically looks like this

enter image description here

Source: https://www.intechopen.com/books/ionic-liquids-new-aspects-for-the-future/tribological-properties-of-ionic-liquids

As you see there is a drop in friction from static to small sliding while still in boundary lubrication. Just focus on the left side of the curve, as the cases with lots of lubricant captured under the contact are on the right side.

Numerically there are different parametrization of the curve done by different software. As you can see below in the examples, the models vary drastically near the speed=0 domain.

MATLAB

Matlab

Source: https://www.mathworks.com/help/physmod/simscape/ref/translationalfriction.html

RECURDYN

RecurDyn

Source: https://functionbay.com/documentation/onlinehelp/default.htm#!Documents/friction.htm

So it is up to you to develop a model for friction that can transition from slipping to stiction and back, and possibly tune it with some testing for the specific materials and conditions you are dealing with.

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