Multi-scale formulations that combine a macro-scale second gradient theory with the classical continuum at the micro-scale are examined in this contribution. In particular, the homogenisation formulations proposed by Kouznetsova et al. (2002), Luscher (2010), Blanco et al. (2016) and Rodrigues Lopes and Andrade Pires (2021) are reviewed and compared. A comprehensive analysis of the underlying microscale constraints for each formulation is undertaken and their impact on the predictive capability is investigated. Remarkably, the formulations proposed by Luscher (2010) and Rodrigues Lopes and Andrade Pires (2021) are equivalent, despite their different initial assumptions. The formulations based on Kouznetsova et al. (2002) may suffer from spurious localisation and mesh dependency in the numerical results at the RVE level as a consequence of the lack of a volumetric constraint. The minimal constraint proposed by Blanco et al. (2016) results in an over constraint that causes a non-physical warping near the RVE corners and edges. Interestingly, an FE2 simulation of the boundary shear layer problem reveals that the macroscopic length scale parameter is independent of the multi-scale formulation. (C) 2021 Elsevier Ltd. All rights reserved.
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