Asymptotic analysis of effective properties of highly concentrated composites

This thesis explores three independent problems, which concern an asymptotic analysis of the overall (or effective or homogenized) properties of high contrast composite materials with inclusions close to touching. In Chapter 2, a two-dimensional mathematical model of a high contrast composite with densely packed inclusions (fibres) which form a periodic array and have an optimal shape (a curvilinear square with rounded-off angles described by a parameter m) is considered. An asymptotic formula for the effective conductivity Adelta of the composite when the interparticle distance delta goes to zero is derived. This formula captures the dependence of Adelta on the parameter m. A rigorous justification for this asymptotic formula is provided by a variational duality approach.;In Chapter 4, a two-dimensional mathematical model of a highly concentrated suspension or a thin film of the rigid inclusions in an incompressible Newtonian fluid is presented. The objectives of this chapter are two-fold: (i) to obtain all singular terms in the asymptotic formula for the overall viscous dissipation rate as the interparticle distance parameter delta tends to zero, (ii) to obtain a qualitative description of a microflow between neighboring inclusions in the suspension. Due to reduced analytical and computational complexity, two-dimensional models are often used for a description of three-dimensional suspensions. The analysis of this chapter provides the limits of validity of two-dimensional models for three-dimensional physical problems and highlights novel features of two-dimensional physical problems (e.g. thin films). It also shows that the Poiseuille type microflow, which had not been taken into account in previous studies, contributes to a singular behavior of the dissipation rate. Examples in which this flow results in anomalous rate of blow up of the dissipation rate in two dimensions are presented. It is also shown that this anomalous blow up has no analog in three dimensions. While previously developed techniques allowed to derive and justify the leading singular term only for special symmetric boundary conditions, a fictitious fluid approach, developed in this chapter, captures all singular terms in the asymptotic formula of the dissipation rate for generic boundary conditions. This approach seems to be an appropriate tool for rigorous analysis of three-dimensional models of suspensions as well as various other models of highly packed composites.;In Chapter 3, a discrete network approximation to the problem of the effective conductivity of a high contrast, densely packed composite in three dimensions is introduced. The inclusions are irregularly (randomly) distributed in a host medium. For this class of arrays of inclusions a discrete network approximation for effective conductivity is derived and a priori error estimates are obtained. A variational duality approach is used to provide rigorous mathematical justification for the approximation and its error estimate.