# Asymptotic analysis of effective properties of highly concentrated composites

Abstract (Summary)

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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 Â? of the composite
when the interparticle distance ? goes to zero is derived. This formula captures the
dependence of Â? on the parameter m. A rigorous justification for this asymptotic
formula is provided by a variational duality approach.
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.
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
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in the asymptotic formula for the overall viscous dissipation rate as the interparticle
distance parameter ? 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.
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School:Pennsylvania State University

School Location:USA - Pennsylvania

Source Type:Master's Thesis

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