  Exact solutions are derived for the problem of a two-dimensional, infinitely
anisotropic, linear-elastic medium containing a periodic lattice of voids. The
matrix material possesses either one infinitely soft, or one infinitely hard
loading direction, which induces localized (singular) field configurations. The
effective elastic moduli are computed as functions of the porosity in each
case. Their dilute expansions feature half-integer powers of the porosity,
which can be correlated to the localized field patterns. Statistical
characterizations of the fields, such as their first moments and their
histograms are provided, with particular emphasis on the singularities of the
latter. The behavior of the system near the void close packing fraction is also
investigated. The results of this work shed light on corresponding results for
strongly nonlinear porous media, which have been obtained recently by means of
the ``second-order'' homogenization method, and where the dilute estimates also
exhibit fractional powers of the porosity.
