This paper is devoted to show a discrete adaptive finite element approximation result for the isotropic two-dimensional Griffith energy arising in fracture mechanics. The problem is addressed in the geometric measure theoretic framework of generalized special functions of bounded deformation which corresponds to the natural energy space for this functional. It is proved to be approximated in the sense of $\Gamma$-convergence by a sequence of discrete integral functionals defined on continuous piecewise affine functions. The main feature of this result is that the mesh is part of the unknown of the problem, and it gives enough flexibility to recover isotropic surface energies.

翻译：本文致力于展示断裂力学中出现的各向同性二维Griffith能量的离散自适应有限元逼近结果。该问题在广义有界变形特殊函数的几何测度论框架下处理，该框架对应于该泛函的自然能量空间。我们证明，该泛函可通过定义在连续分片仿射函数上的一列离散积分泛函在$\Gamma$-收敛意义下逼近。这一结果的主要特征在于网格是问题未知量的一部分，并且具有足够的灵活性来恢复各向同性表面能。