Network approximation in the limit of small interparticle distance of the effective properties of a high-contrast random dispersed composite

We consider a high-contrast two-phase composite such as a ceramic/polymer composite or a fiberglass composite. Our objective is to determine the dependence of the effective conductivity  (or the effective dielectric constant or the effective shear modulus) of the composite on the random locations of the inclusions (ceramic particles or fibers) when the concentration of the inclusions is high. We consider a two-dimensional model and show that the continuum problem can be approximated by a discrete random network (graph). We use variational techniques to provide rigorous mathematical justification for this approximation. In particular, we have shown asymptotic equivalence of the effective constant A for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. We introduce the geometrical interparticle distance parameter using Voronoi tessellation, and emphasize the relevance of this parameter due to the fact that for irregular (non-periodic) geometries it is not uniquely determined by the volume fraction of the inclusions. We use the discrete network to compute  numerically. For this purpose we employ a computer program which generates a random distribution of disks on the plane. Using this distribution we obtain the corresponding discrete network. Furthermore, the computer program provides the distribution of fluxes in the network which is based on Keller's formula for two closely spaced disks. We compute the dependence of  on the volume fraction of the inclusions V for monodispersed composites and obtaine results which are consistent with the percolation theory predictions. For polydispersed composites (random inclusions of two different sizes) the dependence Â(V) is not simple and is determined by the relative volume fraction V_r of large and small particles. We found some special values of V_r for which the effective coefficient is significantly decreased. The computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations.