Γ-convergence for high order phase field fracture: Continuum and isogeometric formulations

We consider high order phase field functionals introduced in Borden et al. (2014) and provide a rigorous proof that these functionals converge to a sharp crack brittle fracture energy. We take into account three dimensional problems in linear elastic fracture mechanics and functionals defined both in Sobolev spaces and in spaces of tensor product B-splines. In the latter convergence holds when the mesh size vanishes faster than the internal length of the phase-field model. On the theoretical level, this condition is natural since the size of the phase field layer, around the crack, itself scales like the internal length; on the numerical level, it should be satisfied by local h-refinement. Technically, convergence holds in the sense of Γ-convergence, with respect to the strong topology of L1, while the sharp crack energy is defined in GSBD2. The constraint on the phase field to take values in [0,1] is taken into account both in the Sobolev setting and in the iso-geometric setting; in the latter, it requires a special treatment since the projection operator on the space of tensor product B-splines is not Lagrangian (i.e., interpolatory).