Contact force-displacement laws and the mechanical behavior of random packs of identical spheres

The contact force-displacement law of two identical elastic spheres can independently display: nonlinear response, path dependence and dissipation due to slip. Omitting relative roll and torsion between the two spheres, a general contact force-displacement law is derived analytically by integrating the differential form of the Hertz-Mindlin solution along the contact displacement path. The Hertz-Mindlin contact law and a different contact law formulated by K, Walton are special cases of this general contact law. Implementation of the contact law in numerical codes may be cumbersome because it requires a full description of the contact load history. Some simplified contact force-displacement laws proposed in the literature that overcome this difficulty are shown to be thermodynamically inconsistent (i.e,, unphysical) since they permit energy generation at no cost. The mean-field approximation and statistical averaging for calculating macroscopic stress-strain relations are discussed with respect to various contact force-displacement laws.