  In this paper, we are concerned with the effective elastic property of a
two-phase high-contrast periodic composite with densely packed inclusions. The
equations of linear elasticity are assumed. We first give a novel proof of the
Flaherty-Keller formula for elliptic inclusions, which improves a recent result
of Kang and Yu (Calc.Var.Partial Differential Equations, 2020). We construct an
auxiliary function consisting of the Keller function and an additional
corrected function depending on the coefficients of Lam\'e system and the
geometry of inclusions, to capture the full singular term of the gradient. On
the other hand, this method allows us to deal with the inclusions of arbitrary
shape, even with zero curvature. An extended Flaherty-Keller formula is proved
for m-convex inclusions, m > 2, curvilinear squares with round off angles,
which minimize the elastic modulus under the same volume fraction of hard
inclusions.
