  The homogenization results obtained by Bacca et al. (Homogenization of
heterogeneous Cauchy-elastic materials leads to Mindlin second-gradient
elasticity. Part I: Closed form expression for the effective higher-order
constitutive tensor. arXiv:1305.2365 Submitted, 2013), to define effective
second-gradient elastic materials from heterogeneous Cauchy elastic solids, are
extended here to the case of phases having non-isotropic tensors of inertia. It
is shown that the nonlocal constitutive tensor for the homogenized material
depends on both the inertia properties of the RVE and the difference between
the effective and the matrix local elastic tensors. Results show that: (i.) a
composite material can be designed to result locally isotropic but nonlocally
orthotropic; (ii.) orthotropic nonlocal effects are introduced when a dilute
distribution of aligned elliptical holes and, in the limit case, of cracks is
homogenized.
