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Cluster method in the theory of fibrous elastic composites

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Consider a 2D multi-phase random composite with different
circular inclusions. A finite number $n$ of inclusions on the infinite plane
forms a cluster. The corresponding boundary value problem for Muskhelishvili's
potentials is reduced to a system of functional equations.

Solution to the functional equations can be obtained by a
method of sucessive approximations or by the Taylor expansion of the unknown
analytic functions.

Next, the local stress-strain fields are calculated and
the averaged elastic constants are obtained in symbolic form. Extensions of
Maxwell's approach and other various self-consisting methods are discussed. An
uncertainty when the number of elements $n$ in a cluster tends to infinity is
analyzed by means of the conditionally convergent series. Basing on the fields
around a finite cluster without clusters interactions one can deduce formulae
for the effective constants only for dilute clusters. The Eisenstein summation
yields new analytical formulae for the effective constants for random 2D
composites with high concentration of inclusions.

This talk is part of the Isaac Newton Institute Seminar Series series.

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