Abstract

The present study is motivated by the fact that homogenization concepts may fail because the scale separation law for different scales does not hold. This violation may exist already in the initial stage or may evolve during material deterioration. A typical example is a long range localized failure by distinct cracks. For this situation the material behaviour of heterogeneous solids on a small scale is projected onto the large or structural scale. We employ a hierarchical two-scale model derived from on a stringent variational concept. It belongs to the superposition based methods with an additive split of the solution u into the large scale ū and small scale part ú. One focus of the present formulation is to allow for locality of the small scale solution within the large scale elements resulting in an efficient solution strategy. At the same time the small scale information exchange over the large scale boundaries is achieved by employing conventional Domain Decomposition concepts and satisfying the related interface constraints. In the current stage of the project the inelastic behaviour of quasi-brittle materials is modeled by continuum damage mechanics in a smeared fashion as well as by the cohesive zone theory for a discrete failure. For the discretization of distinct cracks and material interfaces we apply an extended finite element model (X-FEM) which is supported by level set techniques. It turns out that the X-FEM enhanced small scale solution ú can be perfectly combined with the multi-scale solution scheme. For the time being most of the work is done in a two-dimensional setting; we are currently in the stage to extend the formulation to three-dimensional problems as well.