On a Spring-Network Model and Effective Elastic Moduli of Granular Materials

Two challenges in mechanics of granular media are taken up in this paper: (i) development of adequate numerical discrete element models of topologically disordered granular assemblies, and (ii) calculation of macroscopic elastic moduli of such materials using effective medium theories. Consideration of the first one leads to an adaptation of a spring-network (Kirkwood) model of solid-state physics to disordered systems, which is developed in the context of planar Delaunay networks. The model employs two linear springs: a normal one along an edge connecting two neighboring vertices (grain centers) which accounts for normal interactions between the grains, as well as an angular one which accounts for angle changes between two edges incident onto the same vertex; edges remain straight and grain rotations do not appear. This model is then used to predict elastic moduli of two-phase granular materials—random mixtures of soft and stiff grains —for high coordination numbers. It is found here that an effective Poisson’s ratio, ν^eff, of such a mixture is a convex function of the volume fraction, so that ν^eff may become negative when the individual Poisson’s ratios of both phases are both positive. Additionally, the usefulness of three effective medium theories—perfect disks, symmetric ellipses, and asymmetric ellipses—is tested.