A fully second-order homogenization formulation for the multi-scale modeling of heterogeneous materials

A homogenization-based multi-scale model at finite strains is proposed linking the micro-scale to the macro-scale level through a second-gradient constitutive theory. It is suitable for modeling localization and size effects in multi-phase materials. The macroscopic deformation gradient and second gradient are enforced on a microstructural representative volume element (RVE). The coupling between the scales is defined by means of kinematical insertion and homogenization operators, carefully postulated to ensure kinematical conservation in the scale transition, according to the method of multi-scale virtual power. From the solution of the micro-scale equilibrium, the macroscopic stress tensors are derived based on the principle of multi-scale virtual power. The kinematical constraints at the RVE level are enforced with Lagrange multipliers. A mixed finite element approach is adopted for the numerical solution of the micro and macro second gradient equilibrium problems. It conveniently leads to a formulation where the Lagrange multipliers are the homogenized stresses, and the macroscopic consistent tangent operators for FE2 simulations can be directly derived. The Newton-Raphson scheme is employed to solve the non-linear systems of equations. The numerical results demonstrate the predictive capability of the proposed model. A systematic analysis of the influence of RVE length and the micro-constituent size is performed.