Linear elasticity of planar Delaunay networks. III: Self-consistent approximations

Abstract

Two-phase Delaunay and regular triangular networks, with randomness per vertex, provide generic models of granular media consisting of two types 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. Results of computer simulations of such networks under periodic boundary conditions are used to determine which of several different self-consistent models can provide the best possible approximation to effective Hooke's law. The main objective is to find the effective moduli of a Delaunay 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 effective Poisson's ratio onp for a single-phase Delaunay network with all the spring constantsk assigned according tok=l p. In case of two-phase media, it is found that the Delaunay network is best approximated by a system of ellipses perfectly bonded to a matrix in a symmetric self-consistent formulation, while the regular network is best approximated by a circular inclusion-matrix model. These two models continue to be adequate up to the point of percolation of holes, but the reverse situation of percolation of rigid inclusions is better approximated by the ellipses model in an asymmetric formulation. Additionally, we give results of calculation of Voigt and Reuss bounds of two-dimensional matrix-inclusion composites with springy interfaces.