Effective transport properties of composites with a doubly‐periodic array of fiber pairs and with a triangular array of fibers

Abstract

AbstractThis paper presents a complex variable solution for the effective transport properties of composites with a doubly‐periodic array of fiber pairs. By using the centrosymmetry of the problem, the method of Rayleigh and Natanzon‐Filshtinsky's approach can be simply extended to the problems with two fibers per unit cell. The infinite system constructed in this paper only slightly complicates Rayleigh's system for the problems with one fiber per unit cell. Approximate analytical formulae of the effective transport properties for different fiber‐pair arrays are obtained. The influence of pairwise interaction in fiber pairs on the effective transport properties is discussed in the numerical examples. As a special case of a doubly‐periodic array of fiber pairs, effective transport property of composites with a triangular array of fibers is obtained. The obtained approximate analytical formulae are written in a concise form with good accuracy, thus are convenient for engineering application in most cases, except for those approaching the limit case of percolation when the perfectly conducting fibers become touching. Besides the square array and hexagonal array, the triangular fiber array (similar to carbon atom arrangement in graphene) is another special symmetric fiber array which results into transversely isotropic effective property. Therefore, the present solution for the triangular array is an extension of those for the square array and hexagonal array. The comparison of the results for the three symmetric fiber arrays reveals that the triangular fiber array has the highest conductivity. In addition, accuracy of the present solution is analyzed in the numerical examples.