Mesh and internal length sensitivity of the phase field model for quasi-static and dynamic fracture, comparison with the crack band model

We investigate the mesh sensitivity of the Phase Field Model (PFM), where mesh sensitivity refers to the dependence of quantities of interest – such as total energy dissipation – on the mesh size h. Local damage models are inherently mesh-sensitive because, as h→0, the predicted crack width and energy dissipation vanish. Introducing an internal length scale b mitigates this issue by governing the localization width through b rather than h. Contrary to common assumptions, we show for the first time that the PFM is not universally mesh insensitive for dynamic fracture scenarios. Two mesh refinement strategies are examined: (i) b-fixed, where b remains constant as h decreases, and (ii) F-fixed, where F=b/h is held constant. The PFM retains mesh insensitivity under quasi-static loading with b-fixed refinement. However, at increasing loading rates, damage localization decreases and the damage field becomes more diffuse. In this dynamic limit, the F-fixed approach loses mesh objectivity, and energy dissipation diverges as 1/h, as demonstrated for both 1D fragmentation and 2D crack propagation problems. In contrast, the b-fixed formulation remains mesh insensitive across all loading rates. These findings parallel previously established results for the Crack Band Model (CBM), which effectively treats mesh size as the internal length scale. Finally, by interpreting b as the ratio of surface (toughness) to volume energy dissipation, we provide an energy-based framework to contrast the mesh sensitivity and computational efficiency of PFM and CBM.