  In this contribution, a variational diffuse modeling framework for cracks in
heterogeneous media is presented. A static order parameter smoothly bridges the
discontinuity at material interfaces, while an evolving phase-field captures
the regularized crack. The key novelty is the combination of a strain energy
split with a partial rank-I relaxation in the vicinity of the diffuse
interface. The former is necessary to account for physically meaningful crack
kinematics like crack closure, the latter ensures the mechanical jump
conditions throughout the diffuse region. The model is verified by a
convergence study, where a circular bi-material disc with and without a crack
is subjected to radial loads. For the uncracked case, analytical solutions are
taken as reference. In a second step, the model is applied to crack
propagation, where a meaningful influence on crack branching is observed, that
underlines the necessity of a reasonable homogenization scheme. The presented
model is particularly relevant for the combination of any variational strain
energy split in the fracture phase-field model with a diffuse modeling approach
for material heterogeneities.
