Computational modelling of the mechanical behaviour of metamaterials can be challeng- ing due to intriguing properties and higher-order effects that cannot be predicted by classical continuum mechanics. In this context, the present contribution explores a new multi-scale second-order computational homogenisation formulation at finite strains [1] based on the Method of Multi-scale Virtual Power for investigating such artificial ma- terials with functional properties. The macro-scale is described by a second gradient theory, and the micro-scale Representative Volume Element (RVE) is modelled with the classical Cauchy continuum, where the presence of a void phase is accounted for. The formulation considers the kinematics defined only in the solid domain of the RVE, where a new expression for homogenisation of the second-order gradient is postulated. The robustness and predictive capability of the second-order computational strategy is assessed by multi-scale numerical examples of 2D and 3D architected materials accounting for different deformation modes, encompassing bending, tension/compression-induced undulation, and compression-induced torsion. Comparisons with predictions from first-order homogenisation and Direct Numerical Simulations are conducted. The numerical simulations show that, in general, the second-order strategy is better suited to capture coupling deformation mechanisms and size effects.