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