Relative Transverse Slip and Sliding of Two Spherical Grains in Contact

The analytical models of two spherical grains contact interactions, typical for several classes of slip and sliding regimes in the experimental testing, are proposed. They analyze the cases for coupling or decoupling the frictional microslip and sliding displacements during the kinematically induced sphere translation along a straight trajectory or the force-induced motion from the initially activated contact zone under constant vertical loading. In the slip mode, the evolution of sphere center horizontal displacement obeys the Mindlin-Deresiewicz theory rules either for the force or kinematically induced transverse motions of the sphere. In the frictional sliding mode, it is demonstrated that for the kinematically induced transverse motion of the sphere, the contact tractions are fully governed by the coupled evolution of slip and sliding displacements. When the account for contact slip velocity and the rate of contact plane rotation is made, then the coupling of slip and sliding modes theoretically results in a simple scaling multiplier imposed on the overlap resulted from the sliding mode. It generates a driving force fluctuation and affects the evolution of contact tractions. For transverse sliding of the sphere under constant vertical load and driving force, the contact tractions are essentially governed by the conditions of static equilibrium and are independent of the displacements generated in the slip mode. In this case, the slip displacement provides only the additive term to the sliding displacement of the sphere center, not affecting contact tractions.