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.
Highlights
Multiscale modeling of fracturing solids is a challenging topic that has been the subject of considerable research efforts, cf. the reviews in [1,2]
We model fracturing solids undergoing strain localization using a two-scale approach based on Variationally Consistent Homogenization (VCH)
We propose a continuous-discontinuous homogenization scheme, where the eXtended Finite Element Method (XFEM) is used to represent narrow macroscale localization bands
Summary
Multiscale modeling of fracturing solids is a challenging topic that has been the subject of considerable research efforts, cf. the reviews in [1,2]. A frequently used approach is to perform numerical simulations on Statistical Volume Elements (SVEs), whereby a fundamental challenge is that the SVE looses its representative character upon localization This loss of representativeness leads, for standard first order computational homogenization, to pathological dependence of the numerical results on the macroscale mesh size and the SVE size. We allow for the use of any suitable fracture model (XFEM, interface elements, embedded discontinuities, nonlocal continuum damage) on the microscale, but we consider strong discontinuities on the macroscale, and impose these discontinuities on the SVE in a smeared sense, thereby avoiding the need to explicitly identify and track a localization region in the SVE.
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