On computational homogenization of fracturing continua

Sammanfattning: Ductile fracture is important to control in many industrial processes, be it a desired phenomenon (e.g. in metal cutting) or a failure to be prevented (e.g. in structures subject to blast loading). Increased understanding of the fracture processes can be gained by using computational homogenization, where the nucleation and growth of microscopic cracks is explicitly modeled and included in the effective response of a Statistical Volume Element (SVE). Choosing suitable boundary conditions on the SVE is challenging, because conventional boundary conditions (Dirichlet, Neumann and strong periodic) are inaccurate when cracks are present in the SVE. In the present work, we instead impose periodic boundary conditions in a weak sense on the SVE, leading to a mixed variational format with displacements and boundary tractions as unknowns. By constructing a suitable traction approximation, the boundary conditions can be adapted to the problem at hand in order to gain improved convergence. To this end, we propose a stable traction approximation that is piecewise constant between crack-boundary intersections and we show analytically that the LBB (inf-sup) condition is fulfilled for the proposed approximation. The weakly periodic boundary conditions are combined with the eXtended Finite Element Method (XFEM), cohesive zones and the concept of material forces to perform numerical simulations of materials undergoing crack propagation on the microscale. The numerical examples show that weakly periodic boundary conditions with a suitably chosen traction approximation are more efficient than conventional boundary conditions in terms of convergence with increasing SVE size. This observation holds for stationary cracks as well as for propagating cracks. The work presented in this thesis is concerned with homogenization of damage evolution prior to localization, which is a prerequisite for accurate multiscale modeling of localization.