Formulation and numerical implementation of a variationally consistent multi-scale model based on second-order computational homogenisation at finite strains for quasi-static problems

A multi-scale model which couples a macro-scale strain-gradient theory with a classical continuum at representative volume element (RVE) level is formulated in this contribution for quasi-static problems. It enables modelling the inelastic response of heterogeneous materials at finite strains incorporating their microstructural size and accounting for non-uniform macroscopic deformation fields within the RVE. The development of the multi-scale model is based on the Method of Multi-Scale Virtual Power from which equilibrium equations and homogenisation relations are derived, ensuring a variationally consistent scale transition. The kinematical constraints at the RVE level are enforced with the Lagrange multiplier method and particularised for direct, periodic and minimal conditions. It is shown that the homogenised stresses can be obtained exclusively from the Lagrange multipliers. The finite element solution of the micro and macro equilibrium problems is presented in detail together with the numerical implementation. An efficient strategy is devised to compute the consistent macroscopic tangents required for the FE2 framework. The Newton–Raphson scheme is employed for the iterative solution of the corresponding non-linear system of equations at both scales. Numerical results show the effectiveness of the numerical strategy. A systematic analysis of the influence of the RVE length, the number and size of the micro-constituents, with different constitutive laws at the micro-scale and subjected to several macroscopic deformation states, is performed. The determination of an RVE that is both representative and descriptive of the macroscopic characteristic length is discussed. The accuracy of the numerical results is assessed against a reference direct numerical simulation. •A second-order computational homogenisation model is formulated and implemented.•Lagrange multipliers are used to enforce the microscopic constraints.•Homogenised stresses and consistent tangents are obtained from Lagrange multipliers.•Size effects due to RVE length and constituents’ size are evaluated.•The accuracy of microscopic constraints is assessed against a direct numerical simulation.