Predicting the friction coefficient from multi-asperity contact mechanics
In plain words
Two touching surfaces really meet only at many tiny bumps called asperities. Theory can explain why the true contact area grows in proportion to the load, but it cannot yet predict the value of the friction coefficient for a given pair of materials.
Precise statement
For two elastic or elastoplastic surfaces with self-affine roughness (Hurst exponent H, rms slope), derive the static friction $F = \mu_s N$ with $\mu_s$ independent of apparent area and load, combining the real contact area $A_r(N)$ and the shear strength $\tau_j$ of single junctions. An answer predicts $\mu_s$ from bulk and interfacial material parameters within experimental scatter for specified material pairs.
What would settle it
A first-principles prediction of $\mu_{s}$ for at least two well-characterized material pairs that matches measurements without fitted parameters.
Status in the literature
Unverified note
Proportionality of real contact area to load is established in contact mechanics; prediction of the junction shear strength and hence $\mu_s$ is not (2026).