Computational homogenization of cracked microstructures

Paper i proceeding, 2014

Computational homogenization of microstructures with randomly distributed cracks is studied. The purpose is to facilitate numerical modeling of how cracks on the mesoscale influence the macroscale response. In particular, proper boundary conditions for the Representative Volume Element (RVE) problem as well as a suitable RVE size need to be identified. These investigations will form the basis for our long-term goal to simulate crack propagation on the mesoscale coupled to localization on the macroscale. We perform simulations on many RVEs with randomly distributed and oriented cracks. The area number density and the length of the cracks are kept constant in this example. The extended finite element method (XFEM) is used to model the presence of cracks in the material because it allows different crack patterns to be studied without remeshing. The XFEM implementation is facilitated by a semi-implicit crack description, where the cracks are described as polygon lines that are used to evaluate level set functions. To further improve the robustness of the model, an elastic cohesive zone with a very low stiffness is used as regularization to avoid numerical problems when a part of the domain is completely cut loose by cracks. The model has been implemented in the open source software OOFEM. We investigate the number of RVEs necessary to obtain reliable statistical measures and how the homogenized response depends on the RVE size. These investigations are done with three different boundary conditions: Neumann, Dirichlet with all dofs on the boundary constrained and Dirichlet with continuous dofs constrained but enriched dofs free. The results of the simulations are compared by computing a homogenized stiffness tensor from the solution of the RVE problem. It is found that Neumann boundary conditions give rise to very soft response if a crack cuts the boundary of the RVE in such a way that a part of the structure is cut loose from the rest of the RVE. This is analogous to the case where a hard material with soft inclusions is studied, and an inclusion is located on the boundary. The lower bound for the stiffness, predicted by Neumann boundary conditions, is zero if a single RVE has zero stiffness (i.e. infinite compliance). This is in contrast with Dirichlet boundary conditions, that are less sensitive to soft response in a single RVE. In summary, we find that the choice of boundary conditions has a large influence on the homogenized response in this application.