classical mechanics - 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