Direct micromechanics derivation and DEM confirmation of the elastic moduli of isotropic particulate materials: Part I No particle rotation

We derive the macroscopic elastic moduli of a statistically isotropic particulate aggregate material via the homogenization methods of Voigt (1928) (kinematic hypothesis), Reuss (1929) (static hypothesis), and Hershey (1954) and Krِner (1958) (self-consistent hypothesis), originally developed to treat crystalline materials, from the directionally averaged elastic moduli of three regular cubic packings of uniform spheres. We determine analytical expressions for these macroscopic elastic moduli in terms of the (linearized) elastic inter-particle contact stiffnesses on the microscale under the three homogenization assumptions for the three cubic packings (simple, body-centered, and face-centered), assuming no particle rotation. To test these results and those in the literature, we perform numerical simulations using the discrete element method (DEM) to measure the overall elastic moduli of large samples of randomly packed uniform spheres with constant normal and tangential contact stiffnesses (linear spring model). The beauty of DEM is that simulations can be run with particle rotation either prohibited or unrestrained. In this first part of our two-part series of papers, we perform DEM simulations with particle rotation prohibited, and we compare these results with our theoretical results that assumed no particle rotation. We show that the self-consistent homogenization assumption applied to the locally body-centered cubic (BCC) packing most accurately predicts the measured values of the overall elastic moduli obtained from the DEM simulations, in particular Poissonʹs ratio. Our new analytical self-consistent results lead to significantly better predictions of Poissonʹs ratio than all prior published theoretical results. Moreover, our results are based on a direct micromechanics analysis of specific geometrical packings of uniform spheres, in contrast to all prior theoretical analyses, which were based on difficult-to-verify hypotheses involving overall inter-particle contact distributions. We continue the analysis begun in this first part for the case of unrestrained particle rotation in Part II, Fleischmann et al. (2013).