Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

• Effective second-gradient material is derived from heterogeneous Cauchy elastic solid.
• Nonlocal constitutive tensor for the effective material is given in a closed form.
• A composite material can be designed locally isotropic but nonlocally orthotropic.
• Orthotropic nonlocality is obtained for aligned elliptical holes.

The homogenization results obtained by Bacca et al. (2013a), to identify the 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) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.