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

## TL;DR

This paper proves the $ ext{Gamma}$-convergence of second order phase field functionals, discretized with isogeometric B-splines, to brittle fracture energy, bridging continuum and discrete models in fracture mechanics.

## Contribution

It establishes the $ ext{Gamma}$-convergence of continuum and isogeometric discretized phase field functionals to brittle fracture energy, including a novel construction for the isogeometric setting.

## Key findings

- $ ext{Gamma}$-convergence proven for continuum and discrete models
- Convergence requires mesh size $h$ to be smaller than internal length $ ext{e}$
- Special construction ensures recovery sequences stay within [0,1]

## Abstract

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.