Two-scale modeling of fracturing solids using a smeared macro-to-micro discontinuity transition

Abstract

We consider multiscale modeling of fracturing solids undergoing strain localization, whereby Statistical Volume Elements (SVEs) are used to compute the homogenized macroscopic stresses and the eXtended Finite Element Method (XFEM) is used to represent macroscale displacement discontinuities. These discontinuities are imposed on the localized SVEs in a smeared sense, whereby the smearing width is related to the SVE size and the orientation of the macroscopic discontinuity. This smearing width relation, which is derived within the setting of Variationally Consistent Homogenization (VCH), prevents pathological dependence of the solution on the SVE size. The SVE size insensitivity is further improved by adopting the recently proposed localization aligned weakly periodic boundary conditions. Advantages of the proposed method are that it allows multiscale modeling of localized fracture without restrictive assumptions on the SVE size and without the need to explicitly track a localized region in the SVE.