LINEAR ELASTICITY OF PLANAR DELAUNAY NETWORKS .3. SELF-CONSISTENT APPROXIMATIONS

Two-phase Delaunay and regular triangular networks, with randomness pe r vertex, provide generic models of granular media consisting of two t ypes of grains - soft and stiff. We investigate effective macroscopic moduli of such networks for the whole range of area fractions of both phases and for a very wide range of stiffnesses of both phases. Result s of computer simulations of such networks under periodic boundary con ditions are used to determine which of several different self-consiste nt models can provide the best possible approximation to effective Hoo ke's law. The main objective is to find the effective moduli of a Dela unay network as if it was a field of inclusions, rather than vertices connected by elastic edges, without conducting the computer-intensive calculations of large windows. First, we report on the dependence of e ffective Poisson's ratio on p for a single-phase Delaunay network with all the spring constants k assigned according to k = l(p). In case of two-phase media, it is found that the Delaunay network is best approx imated by a system of ellipses perfectly bonded to a matrix in a symme tric self-consistent formulation, while the regular network is best ap proximated by a circular inclusion-matrix model. These two models cont inue to be adequate up to the point of percolation of holes, but the r everse situation of percolation of rigid inclusions is better approxim ated by the ellipses model in an asymmetric formulation. Additionally, we give results of calculation of Voigt and Reuss bounds of two-dimen sional matrix-inclusion composites with springy interfaces.