Homogenization of heterogeneous Cauchy-elastic materials leads to Mindlin second-gradient elasticity

Through a second-order homogenization procedure, the explicit relation is obtained between the non-local parameters of a second gradient elastic ma- terial and the microstructure of a composite material. This result is instru- mental 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 prob- lems and generalizes earlier findings by Bigoni and Drugan (Analytical deriva- tion of Cosserat moduli via homogenization of heterogeneous elastic materials. J. Appl. Mech., 2007, 74, 741753) 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). From the obtained solution it follows that the equivalent second-gradient Mindlin elastic solid: a) is positive definite only when the discrepancy tensor is negative defined; b) the non-local material symmetries are the same of the discrepancy tensor; c) 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.