c  2015. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/. Computational homogenization of microfractured continua using weakly periodic boundary conditions Erik Svenning ∗ , Martin Fagerstr¨ om, Fredrik Larsson Division of Material and Computational Mechanics Department of Applied Mechanics Chalmers University of Technology Gothenburg, Sweden Abstract Computational homogenization of elastic media with stationary cracks is considered, whereby the macroscale stress is obtained by solving a boundary value problem on a Statistical Volume Element (SVE) and the cracks are rep- resented by means of the eXtended Finite Element Method (XFEM). With the presence of cracks on the microscale, conventional BCs (Dirichlet, Neu- mann, strong periodic) perform poorly, in particular when cracks intersect the SVE boundary. As a remedy, we herein propose to use a mixed vari- ational format to impose periodic boundary conditions in a weak sense on the SVE. Within this framework, we develop a novel traction approximation that is suitable when cracks intersect the SVE boundary. Our main result is the proposition of a stable traction approximation that is piecewise constant between crack-boundary intersections. In particular, we prove analytically that the proposed approximation is stable in terms of the LBB (inf-sup) condition and illustrate the stability properties with a numerical example. We emphasize that the stability analysis is carried out within the setting of weakly periodic boundary conditions, but it also applies to other mixed problems with similar structure, e.g. contact problems. The numerical ex- * Corresponding author. Email address: erik.svenning@chalmers.se (Erik Svenning) Preprint submitted to Comput. Methods Appl. Mech. Engrg. November 18, 2015