Elliptic functions and lattice sums for effective properties of heterogeneous materials

Abstract

Effective properties of fiber-reinforced composites can be estimated by applying the asymptotic homogenization method. Analytical solutions are possible for infinite long circular fibers based on the elliptic quasi-periodic Weierstrass Zeta function. This process leads to numerical convergences issues related to lattice sums calculations. The lattice sums original series converge slowly, which make the calculation difficult. This problem needs to be addressed because effective properties are highly sensitive to these values. Therefore, a systematic review and analysis for the lattice sums are a necessity. In the present work, the Eisenstein–Rayleigh lattices sums are reviewed and numerically implemented for fiber-reinforced composites with parallelogram unit periodic cell whose fibers are centered, or not, at the coordinate origin. Numerical values are reported and compared with available data in the literature obtaining good agreements. In this work, new Eisenstein–Rayleigh lattice sums are obtained that are easy to implement and a set of tables with numerical values are given.