Constructive Homogenization Of Dispersed Random Composites. Part I. Theory

We consider a two-dimensional, two-component dispersed random composite consisting of non-overlapping, identical circular disks embedded in a uniform host. The homogenization theory imposes the condition of a strictly stationary distribution of inclusions. The aRVE theory addresses the constructive question of homogenization. The Eisenstein summation method is used to analyze the convergence of the series that arise during the homogenization process. These series are defined for a random set of inclusion centers. Theoretical investigation of their convergence is necessary for analytical derivation and computational simulations of the effective constants. It is demonstrated that the coefficients of the power concentration series of the effective conductivity tensor are expressed in terms of conditionally convergent series. Various methods of summation formally yield different results. Their proper interpretation is discussed.