Linear elasticity of planar delaunay networks. Part II: Voigt and Reuss bounds, and modification for centroids

The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation - a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.