Tearing a neo-Hookean sheet. Part I: insights into the crack tip fields by a phase field damage model

In this paper, we investigate the effects of material damage on mode I crack tip fields near an advancing crack in a neo-Hookean sheet using a phase field model. Phase-field governing equations are introduced and solved using the finite element method combined with a Riks path-following approach, which allows for tracking of non-monotonic evolution of the equilibrium path in a fracture process. We calibrate the scale parameter in the phase field model based on an energy method where the prescribed fracture energy is equal to the real energy dissipation in fracture of a 1D bar. We observe good agreement between the stress-stretch relation of the phase field model and experimental data in tearing of a long notched neo-Hookean strip in multiple loading cases. We find that the crack tip fields are self similar as the crack propagates. Due to material damage, the normal stress component shows a peak value ahead of the crack rather than the singular behavior observed in a purely elastic neo-Hookean material. We show that the magnitude of the peak stress is proportional to the critical fracture energy and inversely proportional to the internal length scale in a scaled coordinate system.