Elastic moduli of Delaunay Networks: Exact results versus effective medium theories

Determination of effective moduli of granular media with random internal geometry and of multiphase composition is one of the important problems in mechanics of granular media. Here we study it in a two-dimensional setting using Delaunay networks (dual to Voronoi planar tessellations) with linear elastic central-force interactions). We assume the Delaunay vertices (i.e. Voronoi cells) to be occupied randomly by either a hard phase or a soft phase. First, we compute effective moduli of such networks using very large networks under periodic boundary conditions. These exact results are then compared to moduli one would obtain from various effective medium theories.