title: Closed-form evaluation of two-dimensional static lattice sums.
abstract: Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sumsS2(for conductivity) andT2(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 betweenS2andT2for different lattices are established. In particular, the valueS2=πfor the square and hexagonal arrays is discussed andT2=π/2 for the hexagonal is deduced.
authors.affiliation: Department of Mathematics, Faculty of Sciences , University of Porto , Campo Alegre Street, 687, 4169-007 Porto, Portugal.
authors.forename: S
authors.initials: S
authors.lastname: Yakubovich
authors.affiliation: Department of Differential Equations and Statistics, Faculty of Mathematics and Natural Sciences , University of Rzeszow , Pigonia 1, 35-959 Rzeszow, Poland.
authors.forename: P
authors.initials: P
authors.lastname: Drygas
authors.affiliation: Institute of Computer Sciences , Pedagogical University , ul.Podchorazych 2, Krakow 30-084, Poland.
authors.forename: V
authors.identifier: 0000-0001-6963-1896
authors.initials: V
authors.lastname: Mityushev
keywords: Eisenstein summation method; Mellin transform; Riemann’s zeta-function; complete elliptic integrals; effective properties of two-dimensional composites; lattice sum