  It is shown that second-order homogenization of a Cauchy-elastic dilute
suspension of randomly distributed inclusions yields an equivalent second
gradient (Mindlin) elastic material. This result is valid for both plane and
three-dimensional problems and extends earlier findings by Bigoni and Drugan
(Analytical derivation of Cosserat moduli via homogenization of heterogeneous
elastic materials. J. Appl. Mech., 2007, 74, 741-753) from several points of
view: (i.) the result holds for anisotropic phases with spherical or circular
ellipsoid of inertia; (ii.) the displacement boundary conditions considered in
the homogenization procedure is independent of the characteristics of the
material; (iii.) a perfect energy match is found between heterogeneous and
equivalent materials (instead of an optimal bound). The constitutive
higher-order tensor defining the equivalent Mindlin solid is given in a
surprisingly simple formula. Applications, treatment of material symmetries and
positive definiteness of the effective higher-order constitutive tensor are
deferred to Part II of the present article.
