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In this paper a second-order homogenization approach for periodic material isderived from an appropriate representation of the down-scaling that correlatesthe microdisplacement field to the macro-displacement field and themacro-strain tensors involving unknown perturbation functions. These functionstake into account of the effects of the heterogeneities and are obtained by thesolution of properly defined recursive cell problems. Moreover, theperturbation functions and therefore the micro-displacement fields result to besufficiently regular to guarantee the anti-periodicity of the traction on theperiodic unit cell. A generalization of the macro-homogeneity condition isobtained through an asymptotic expansion of the mean strain energy at themicro-scale in terms of the microstructural characteristic size e; the obtainedoverall elastic moduli result to be not affected by the choice of periodiccell. The coupling between the macro- and microstress tensor in the periodiccell is deduced from an application of the generalised macrohomogeneitycondition applied to a representative portion of the heterogeneous material(cluster of periodic cell). The correlation between the proposed asymptotichomogenization approach and the computational second-order homogenizationmethods is obtained through an approximation of the macrodisplacement fieldbased on a second-order Taylor expansion. The form of the overall elasticmoduli obtained through the two homogenization approaches, here proposed, isanalyzed and the differences are highlighted.

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits