Analytic asymptotic formulas for effective parameters of planar elastic composites

We investigate the effective elastic properties of periodic dilute two-phase composites consisting of an homogeneous isotropic matrix and a periodic array of rigid inclusions. We assume the rigid inclusion in a unit cell is a simply connected, bounded domain so that there exists an exterior conformal mapping corresponding the inclusion. Recently, an analytical series solution method for the elastic problem with a rigid inclusion was developed based on the layer potential technique and the geometric function theory \cite{Mattei:2021:EAS}. In this paper, by using the series solution method, we derive expression formulas for the elastic moment tensors--the coefficients of the multipole expansion associated with an elastic inclusion--of an inclusion of arbitrary shape. These formulas for the elastic moment tensors lead us to analytic asymptotic formulas for the effective parameters of the periodic elastic composites with rigid inclusions in terms of the associated exterior conformal mapping.