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

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