On a volume averaged measure of macroscopic reinforcement slip in two‐scale modeling of reinforced concrete

A two-scale model for reinforced concrete, in which the large-scale problem formulation is enriched by an effective reinforcement slip variable, is derived from the single-scale model describing the response of plain concrete, reinforcement steel, as well as the bond between them. The subscale problem on the Representative Volume Element (RVE) is correspondingly defined as finding the response of the RVE subjected to effective variables (strain, slip, and slip gradient) imposed from the large-scale.  A novel volumetric definition of effective reinforcement slip and its gradient is devised, and the corresponding subscale problem is formulated.  The newly-defined effective variables are imposed on the RVE in a weak sense via Lagrange multipliers. The response of the RVEs of different sizes was investigated by means of pull-through tests, and the novel boundary condition type was used in FE^2 analyses of a deep beam. Locally, prescribing the macroscopic reinforcement slip and its gradient in the proposed manner resulted in reduced RVE-size dependency of effective work conjugates, which allows for more objective description of reinforcement slip in two-scale modelling of reinforced concrete. Globally, this formulation produced more consistent amplitudes of effective slip fluctuations, as well as more consistent maximum crack width predictions.