  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.
