A dispersive multi-scale crack model for quasi-brittle heterogeneous materials under impact loading

A dispersive multi-scale model is presented to model failure in heterogeneous quasi-brittle materials under high frequency loading conditions. In the dispersive multi-scale model, the heterogeneous model undergoing localized failure is replaced by a homogeneous macro-scale model with a cohesive crack and a meso-scale model with diffuse damage. Each material point of the macro-scale model is linked to a heterogeneous meso-scale model. The macro-crack is modeled as a strong discontinuity and the gradient-enhanced damage model is used to model diffuse damage in the meso-scale model. The constitutive law for the bulk material is obtained from the meso-scale model analysis using a standard computational homogenization scheme. The cohesive law for the macro-crack is obtained using a continuous-discontinuous homogenization scheme which is based on a failure zone averaging technique. In the dispersive multi-scale model, at the macro-scale, a dynamic analysis is performed and the meso-scale model is solved as a quasi-static problem. The meso-scale inertia forces are taken into account via a dispersion tensor which only depends on the meso-scale model material properties and the heterogeneity of the material. The meso-scale inertia effects appear as additional body forces in the macro-scale model and cause dispersion of the propagating wave. The effect of dispersion on the cohesive cracking is captured via a rate dependent cohesive law. The dispersive multi-scale model is verified against a direct numerical simulation and the objectivity of the scheme with respect to the representative volume element size is shown.