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Advances in Mathematical Physics
Volume 2015, Article ID 535128, 15 pages
Research Article

Random 2D Composites and the Generalized Method of Schwarz

Pedagogical University, ul. Podchorazych 2, 30-084 Krakow, Poland

Received 3 September 2015; Accepted 26 November 2015

Academic Editor: Xavier Leoncini

Copyright © 2015 Vladimir Mityushev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Two-phase composites with nonoverlapping inclusions randomly embedded in matrix are investigated. A straightforward approach is applied to estimate the effective properties of random 2D composites. First, deterministic boundary value problems are solved for all locations of inclusions, that is, for all events of the considered probabilistic space by the generalized method of Schwarz. Second, the effective properties are calculated in analytical form and averaged over . This method is related to the traditional method based on the average probabilistic values involving the -point correlation functions. However, we avoid computation of the correlation functions and compute their weighted moments of high orders by an indirect method which does not address the correlation functions. The effective properties are exactly expressed through these moments. It is proved that the generalized method of Schwarz converges for an arbitrary multiply connected doubly periodic domain and for an arbitrary contrast parameter. The proposed method yields an algorithm which can be applied with symbolic computations. The Torquato-Milton parameter is exactly written for circular inclusions.