what is the relation between friction, clearance/tolerance, roughness and mechanical properties in a prismatic joint?

Consider an axisymmetric model of a prismatic joint:

Assumptions:

• The circular shaft moves linearly, only in longitudinal/axial direction
• The shaft has a diameter of $D$
• The clearance between shaft and the bearing is $c$.
• Mechanical properties of shaft and bearing are known (e.g. Young's modulus $E$, Poisson's ratio $\nu$, yield stress $\sigma_y$, hardness...)
• Tribological properties of the surfaces are known (e.g. arithmetical mean deviation roughness $R_a$)
• There is a axial dry friction force between shaft and the bearing (e.g. Coulomb friction )
• There are no lateral/radial forces

What I know:

• there is an inverse relation between friction force and clearance $F_f\sim\frac{1}{c}$ and there is a maximum clearance $c_{max}$ where the friction becomes practically zero

• friction and roughness have a complicated relationship! As I have understood there can be multiple minimas within a certain range:

• friction and elasticity also have an inverse relationship $F_f\sim\frac{1}{E}$ so the softer the material the lower the friction should be (or not?)
• the lower the hardness and yield stress the easier the materials worn so the friction should be lower

What I need:

A mathematical formulation based on the physics of the problem so I can fit to my experimental observations. I'm looking for an equation like $F_f \approx F\left(R_a \, , \, c \, , \, E \, , \, ... \right)$ which relates friction to mechanical properties of the materials, Tribological properties of the contact surfaces and the geometrical clearance.

P.S.1. I wasn't sure if I should ask this question here on in engineering

P.S.2. I miss some tags here contact mechanics, wear, tribology, micromechanics...

P.S.3. These are the attempts I'm taking to solve the problem myself:

1. finding the relationship between normal pressure and tolerance here
2. finding the relationship between roughness and displacement here
• Comments are not for extended discussion; this (very interesting) conversation has been moved to chat. – rob Mar 1 '18 at 16:42
• @rob thanks but I try to avoid chats because as far as I know they will not be indexed by the search engines. besides the mathjax equations are not shown properly – Foad Mar 1 '18 at 16:51
• Comments aren't indexed by SE's internal search, while chat transcripts are searchable both within SE and by external search engines. But if you want stuff to hang around, you should use comments (or chat) as a staging area to decide how to edit your post, and you should be unsurprised if your comments disappear. People in Physics Chat can help you get MathJax working in your other chat rooms. – rob Mar 1 '18 at 19:57
• Engineering would probably be the preferred site. Check out P70: diva-portal.org/smash/get/diva2:205807/FULLTEXT01.pdf and t.tribologia.eu/trib/artykul/… in particular, for which assumptions should be made. There are many variables that depend upon many things, take for example hardness (and of which surface), and surface finish; too smooth and the surfaces stick (like microscope slides). ... – Rob Mar 4 '18 at 16:21
• @Foad - Glad to be a bit of help. You should add the link from your comment into your question, so people can see what has been done and don't duplicate their efforts (and fix the "orn" spellcheck error). You can see from the answer there (which doesn't answer this question) how involved this is. – Rob Mar 5 '18 at 14:21