A rate-dependent multi-scale crack model for concrete

A multi-scale numerical approach for modeling cracking in heterogeneous quasi-brittle materials under dynamic loading is presented. In the model, a discontinuous crack model is used at macro-scale to simulate fracture and a gradient-enhanced damage model has been used at meso-scale to simulate diffuse damage. The traction-separation law for the cohesive zone model at macro-scale is obtained from the meso-scale information through the discontinuous computational homogenization method. The method is based on the so-called failure zone averaging scheme in which the averaging theorem is used over the active damaged zone of the meso-scale. Objectivity with respect to the local-scale sample size in the softening regime is obtained in this fashion. In order to evaluate the macroscopic traction at each integration point on the crack, at each time step of the macro model solution, a static boundary value problem is solved for the representative volume element (RVE) whose size is much smaller than the macro length-scale and the macroscopic wave-length. The effect of the crack opening rate on the macro cohesive law is taken into account by relating the material properties of the meso-scale model to the macro crack opening rate. The objectivity of the model response with respect to the representative volume element (RVE) size is demonstrated for wave propagation problems. The rate-dependent multi-scale model is then verified by comparison with a direct numerical simulation (DNS).