Through a second-order homogenization procedure, the explicit relation is obtained between the non-local parameters of a second gradient elastic material and the microstructure of a composite material. This result is instrumental for the definition of higher-order models, to be used for the analysis of mechanics at micro- and nano-scale, where size-effects become important.

The obtained relation is valid for both plane and three-dimensional problems and generalizes earlier findings by Bigoni and Drugan (Analytical derivation of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech., 2007, 74, 741753) from several points of view:​

1. the result holds for anisotropic phases with spherical or circular ellipsoid of inertia;​
2. the displacement boundary conditions considered in the homogenization procedure is independent of the characteristics of the material;​
3. a perfect energy match is found between heterogeneous and equivalent materials (instead of an optimal bound).

From the obtained solution it follows that the equivalent second-gradient Mindlin elastic solid:​

1. is positive definite only when the discrepancy tensor is negative defined;​
2. the non-local material symmetries are the same of the discrepancy tensor;​
3. the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response.

Finally, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in particular cases.