Unlocking the Potential of Second-order Computational Homogenisation: An Overview of Distinct Formulations and a Guide for their Implementation

Multi-scale models enriched with the second gradient of the displacement field and based on the concept of representative volume element (RVE) are examined and discussed in detail in this paper. The development of new models relies on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived employing variational arguments. Two classes of enhanced multi-scale constitutive models are analysed: standard second-order homogenisation and fully second-order homogenisation. On the computational side, the numerical treatment of second gradient equilibrium problems with a mixed approach is addressed systematically within an implicit finite element discretisation in conjunction with the full Newton–Raphson scheme for the iterative solution of the corresponding non-linear systems of equations. Different formulations available in the literature for standard second-order homogenisation are presented, along with a critical appraisal of their assumptions and limitations. A macroscopic second-gradient constitutive law with an orthotropic length scale is developed based on numerical findings. The impact of finite strains, the micro constituents’ behaviour, and the RVE morphology on the resulting macroscopic length scale parameter’s evolution is assessed. A fully second-order homogenisation-based model at finite strains is also presented. The suitability of this class of models to capture size effects due to the micro-constituent’s size is illustrated by an FE 2 simulation.