Computational homogenization of microfractured continua using weakly periodic boundary conditions

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 represented by means of the eXtended Finite Element Method (XFEM). With the presence of cracks on the microscale, conventional BCs (Dirichlet, Neumann, strong periodic) perform poorly, in particular when cracks intersect the SVE boundary. As a remedy, we herein propose to use a mixed variational 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 examples show that the proposed traction approximation is more efficient than conventional boundary conditions (Dirichlet, Neumann, strong periodic) in terms of convergence with increasing SVE size