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Starting from a Cauchy elastic composite with a dilute suspension of randomlydistributed inclusions and characterized at first-order by a certaindiscrepancy tensor (see part I of the present article), it is shown that theequivalent second-gradient Mindlin elastic solid: (i.) is positive definiteonly when the discrepancy tensor is negative defined; (ii.) the non-localmaterial symmetries are the same of the discrepancy tensor, and (iii.) thenonlocal effective behaviour is affected by the shape of the RVE, which doesnot influence the first-order homogenized response. Furthermore, explicitderivations of non-local parameters from heterogeneous Cauchy elasticcomposites are obtained in the particular cases of: (a) circular cylindricaland spherical isotropic inclusions embedded in an isotropic matrix, (b)n-polygonal cylindrical voids in an isotropic matrix, and (c) circularcylindrical voids in an orthortropic matrix.

international journal of solids and structures

Mindlin second-gradient elastic properties from dilute two-phase  Cauchy-elastic composites Part II: Higher-order constitutive properties and  application cases