PFM for crack nucleation and propagation in isotropic heterogeneous materials

We address herein the -version of the phase field method (-PFM) applied to linear elastic heterogeneous materials with continuous variation of Young modulus and fracture toughness GIc along crack propagation and crack initiation directions. The method is investigated through both finite element simulations and semi-analytical formulations. In crack propagation scenarios, we show that using a naive GIc correction yields results in close agreement with classical Fracture Mechanics predictions, within ± 3 %, and that energy convergence is achieved as the regularization length ₀ decreases, indicating potential Γ -convergence. For crack nucleation, we derive semi-analytical pseudo-homogeneous damage profiles in heterogeneous domains and validate them against finite element results. We find that damage evolves until an ultimate strain beyond which a localized crack forms orthogonally to the material variation, marking a bifurcation from the initial solution. Numerical stability is assessed through the second Gateaux derivative, and the continuation algorithm is used to track the post-instability path. These findings advance the understanding of crack nucleation and propagation in heterogeneous materials and offer a validated modeling framework for future fracture prediction with PFM in Functionally-Graded Materials and human long bones.