Rough elastic spheres in contact: Memory effects and the transverse force

Abstract

A pair of identical elastic spheres is simultaneously pressed against one another and sheared sideways causing both normal and tangential relative displacement of their centers. The contacting surfaces are assumed to be rough and the contact zone is modeled using the theories of Hertz and Mindlin. Because we focus our attention on the case of perfect adhesion, we are able, for the first time, to derive a rule for the variation in tangential force T as a function of the displacement components. T displays a memory effect through its dependence upon a, the radius of the contact zone which is uniquely related to the normal displacement, and s, the tangential displacement. Thus, ΔT = C t aΔs if a is not decreasing, and C t is a material parameter. Otherwise, ΔT = C t Δ( as) − C t s u Δa where s u is the value of s when the contact zone w most recently at the same radius. The consequences for work and energy are explored in detail. We derive a simple analytical formula for the work done in going around an arbitrary closed path. Applications to acoustic wave propagation in a granular medium are discussed.