TY - JOUR

T1 - Generalized Clausius-Mossotti formula for random composite with circular fibers

AU - Berlyand, L.

AU - Mityushev, V.

N1 - Funding Information:
The work of L. Berlyand was supported by NSF Grant DMS-9971999. Part of this work was done when V. Mityushev visited L. Berlyand at the University of Akron. He is grateful for the hospitality and support of his visit. We are grateful to G. Milton for useful discussions on the analyticity of the effective conductivity as a function of the volume fraction. We are also grateful to O. Bruno for bringing our attention to the extremal property in the dilute limit in ref. 15.

PY - 2001/1

Y1 - 2001/1

N2 - An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius-Mossotti (Maxwell-Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays ("shaking" geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

AB - An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius-Mossotti (Maxwell-Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays ("shaking" geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

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U2 - 10.1023/A:1026512725967

DO - 10.1023/A:1026512725967

M3 - Article

AN - SCOPUS:0035530273

VL - 102

SP - 115

EP - 145

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1-2

ER -