On the micro-to-macro transition of reinforcement slip in two-scale modelling

A two-scale model for reinforced concrete, in which the macroscopic problem formulation is enriched by an effective reinforcement slip variable is considered. The corresponding subscale problem on the Representative Volume Element (RVE) is defined in terms of finding the response of the RVE subjected to effective variables (strain, slip, slip gradient) imposed from the macroscale. In this contribution, the two possible approaches of prescribing the effective reinforcement slip are discussed. Namely, a boundary definition of the macroscopic slip can be employed and the variable is thus prescribed only at boundary of the RVE, which corresponds to Dirichlet boundary conditions. Alternatively, a volumetric averaging measure can be used to define the effective reinforcement slip. In this case, the effective variables are imposed on the RVE in a weak sense via Lagrange multipliers. It is shown that the weak enforcement of reinforcement slip and its gradient resulted in objective interpretation of the effective variable (and its work conjugates), which was not pathologically dependent on the size of the RVE.