The incremental response of random aggregates of identical round particles

This paper is concerned with a dense, randomly packed, granular material that consists of identical spheres or disks with elastic, frictional interactions, that is first isotropically compressed and subsequently loaded along an arbitrary stress path. An analytical relationship between the overall stress and strain increments is determined for the pre-failure regime. The purpose of the modelling is to understand how this relation depends upon the features of the packing and the particle interactions. From the outset it is recognised that the packing and interactive properties for these materials may vary substantially from grain to grain and the heterogeneity introduced in this manner is fully accounted for. Moment equilibrium equations are solved for each particle and force equilibrium equations are solved for each neighbourhood. Then, the heterogeneity of the aggregate is taken into account by introducing means and fluctuations in the description of the local deformations and the measures of the particles and interactions. The general development is illustrated with an example in two dimensions in which the packing and contact interactions are approximated by angular distributions and the heterogeneity is introduced by variations in these. For an isotropic medium with constant contact stiffnesses the theory provides predictions that compare well with results obtained from numerical simulations.