  The overall behavior of a 2D lattice of voids embedded in an anisotropic
matrix is investigated in the limit of vanishing porosity f. An
effective-medium model (of the Hashin-Shtrikman type) which accounts for
elastic interactions between neighboring voids, is compared to Fast Fourier
Transform numerical solutions and, in the limits of infinite anisotropy, to
exact results. A cross-over between regular and singular dilute regimes is
found, driven by a characteristic length which depends on f and on the
anisotropy strength. The singular regime, where the leading dilute correction
to the elastic moduli is an O(f^{1/2}), is related to strain localization and
to change in character - from elliptic to hyperbolic - of the governing
equations.
