Approximation of fracture energies with $p$-growth via piecewise affine finite elements

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSBD^p(Omega)$, $pin(1,infty)$, their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of $GSBD^p$ {functions}, {for $pin(1,infty)$}, with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of $C^1$ hypersurfaces. The strains of the approximating functions converge strongly {in $L^p$ to the strain of the target,} and the area of the{ir jump sets converge to the area of the target}. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the {Freudenthal} partition of a cubic grid.