$\Gamma$-convergence for high order phase field fracture: continuum and isogeometric formulations

We consider second order phase field functionals, in the continuum setting, and their discretization with isogeometric tensor product B-splines. We prove that these functionals, continuum and discrete, $\Gamma$-converge to a brittle fracture energy, defined in the space $GSBD^2$. In particular, in the isogeometric setting, since the projection operator is not Lagrangian (i.e., interpolatory) a special construction is needed in order to guarantee that recovery sequences take values in $[0,1]$; convergence holds, as expected, if $h = o (\varepsilon)$, being $h$ the size of the physical mesh and $\varepsilon$ the internal length in the phase field energy.