Friction forces on a solid experiencing more than one contact

A massive, macroscopic object can be in equilibrium under its own weight plus reaction forces from a number of distinct macroscopic contacts A, B, C, etc. The reaction forces cannot be obtained from the classical friction coefficients. For instance, a weight G slides on a tilted, planar support, and stops. Assume that the sliding surface of G is made of two macroscopic regions A and B, with different friction features. What is the distribution of tangential forces on A and B? We study this using the model of dry friction analysed by Caroli and Nozieres (1996), with independent asperities leading to multistable equilibria. The stopping position involves a certain amount of recoil delta, which is derived from a geometric construction on areas. Knowing the relation between delta and the overall tangential force F-A + F-B = F, we can construct the partition coefficients F-A/F and F-B/F. They can be positive or negative, and they are not constant when F varies. These partition coefficients are conceptually needed for more complex problems, such as the role of walls for the stress distribution in a silo filled with grain.