Closed-form evaluation of two-dimensional static lattice sums

Abstract

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S 2 (for conductivity) and T 2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S 2 and T 2 for different lattices are established. In particular, the value S 2 = π for the square and hexagonal arrays is discussed and T 2 = π /2 for the hexagonal is deduced.