The length scale dependency of phase field method: and comparison with the crack band model

In continuum damage models involving strain softening, various Quantities of Interest (QoIs) such as energy dissipation are pathologically dependent on the discretization resolution (i.e. mesh size). For finite element methods, this is referred to as the mesh size sensitivity, implying lack of regularization. The crack band model addresses this issue by adjusting the slope of the softening part of strain-stress response, as a function of the mesh size, to ensure the same fracture energy regardless of mesh size. It was however recently shown that this remedy only works for quasi-static problems and for dynamic problems the correct approach is to “do nothing”, that is, not to adjust this slope [1].
Another approach to address the mesh bias of bulk damage models is to introduce an internal length scale b in the fracture model, as in gradient damage and the Phase Field (PF) models. The sensitivity of fracture QoIs to b is referred to as the internal length scale dependence. While as expected all PF models do not exhibit mesh sensitivity, only the Cohesive Zone Model (CZM) based PF models (CZM-PF) are shown to be internal-length scale independent for quasi-static fracture [2].
We investigate the mesh and internal length scale sensitivity of PF model for quasi-static and dynamic fracture problems. We show that the internal length scale dependency of the CZM-based PF method does not extend to dynamic fracture at sufficiently high rates. In fact, like the crack band model the correct approach is to “do nothing”, that is in the context of PF models not to adjust b based on element size. Results will be presented from 0D analytical models to 1D and 2D simulations.
[1] T Abdullah, and K Kirane. Strain rate dependence of the mesh objectivity in dynamic fracture analyses with the crack band model. EFM 269: 108501, 2022
[2] TK Mandal, VP Nguyen, and J-Y Wu. Length scale and mesh bias sensitivity of phase-field models for brittle and cohesive fracture. EFM 217: 106532, 2019