EFFECTIVE PROPERTIES OF FIBROUS COMPOSITES AND CLUSTER CONVERGENCE

We study the mechanical properties of unidirectional fibers with circular section embedded into a matrix by the method of complex potentials. First, a finite number n of inclusions is considered and the corresponding local elastic fields are determined. Next, the limit local fields and the effective elastic constants are estimated as n -> infinity. Such a cluster method can be considered as an extension of Maxwell's approach from single-to n-inclusions problems. For many years it was thought that Maxwell's and Clausius-Mossotti's approximations can be improved by taking into account interactions between pairs of spheres, triplets of spheres, and so on. However, it was recently demonstrated that the field around a finite cluster of inclusions can yield a formula for the effective conductivity only for dilute clusters. The proper treatment of the cluster method is equivalent to the proper limit n -> infinity leading to a conditionally convergent integral (series). Such an integral was estimated due to Batchelor, Jeffrey, O'Brein, and others. It was based on the physical intuition rather than a rigorous mathematical investigation. The present paper is devoted to the cluster method for two dimensional (2D) elasticity problems. It is established that the conditionally convergent series used for the local fields and for the effective properties which have to be defined by different methods of summation called for definiteness of the Eisenstein-Rayleigh and Maxwell approaches. The first approach is based on the Eisenstein summation method applied by Rayleigh. The second approach corresponds to Maxwell's self-consistent method and it is based on the symmetric summation. It is surprising that the Eisenstein-Rayleigh and Maxwell approaches yield different values for the conditionally convergent series. This requires a subtle application of the cluster method to computation of the effective constants. This paper explains the different limit values of series and shows how to apply the cluster method to 2D elastic composites.