Micro-Mechanical modelling of load dependent friction in contacts of elastic spheres

Abstract

Recent experimental measurements using twin-axis load cells in storage vessels for bulk solids have revealed load-dependent variation of the wall friction angles measured in the presence of small normal loads on smooth silo walls. The effect is quite appreciable with smooth particles such as plastic beads and glass ballotini, but particles with rough surfaces, like some agricultural seeds, show no such effect.The inclusion of an intrinsic shear resistance at zero normal load into the frictional behaviour (after Briscoe and Tabor) provides a satisfactory prediction of tangential forces required to cause gross sliding of smooth particles over smooth silo walls. This work applies the same adhesive friction concepts to the microdisplacements that occur before full tangential sliding develops.Mindlin analysed the traction profiles for two spheres in contact assuming a constant coefficient of friction, independent of normal stress inside the contact area, with microslip occurring in those portions of the contact area that had a tangential-to-normal stress ratio exceeding the friction coefficient. The assumptions of that analysis preclude the existence of the observed load dependence of wall friction angles. In this paper we re-examine the micromechanical nature of contacts between elastic spheres incorporating tangential stress limits that contain two terms: one proportional to the normal stress at each point in the contact area, the other independent of normal stress. Two possible physical (and mathematical) interpretations of the stress-independent term are discussed, one called adhesive slip and the other stick slip. The resulting traction profiles are examined to determine implications for both microslip and gross-sliding frictional behaviour. A complete theoretical analysis requires the solution of the surface displacement integral over the contact region for a chosen form of the tangential traction profile satisfying appropriate boundary conditions for elastic displacements. Upper and lower bounds and an approximate form for the resulting tangential force-displacement curves are developed and it is shown that such models result in a load-dependent friction behaviour that is very similar to recent experimental wall friction and individual particle friction measurements.