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Application of Volterra-Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. I. General Procedure

Konstantin Z. Markov
SIAM Journal on Applied Mathematics
Vol. 47, No. 4 (Aug., 1987), pp. 831-849
Stable URL: http://www.jstor.org/stable/2101571
Page Count: 19
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Application of Volterra-Wiener Series for Bounding the Overall Conductivity of Heterogeneous Media. I. General Procedure
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Abstract

A general method of placing optimal bounds on the overall conductivity of random heterogeneous media is proposed. It utilizes truncated Volterra-Wiener functional series, generated by the random conductivity field κ (x), as classes of trial functions for the classical variational principles. The method simplifies, unifies and/or generalizes the earlier proposed variational techniques of Beran [3], Dederichs and Zeller [10], Hori [15], Kröner [17] and Prager [25]. The general procedure is displayed in detail in the simplest case of interest, the construction of the optimal third-order bounds, that requires knowledge of the two- and three-point correlation functions for κ (x). The evaluation of these bounds is reduced to the solution of an integrodifferential equation whose coefficients and kernels are expressed through the said correlation functions. A perturbation solution to this equation is given. For Miller's cell materials the equation is explicitly solved and the obtained optimal bounds are shown to coincide with those of Beran-Miller.

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