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

## TL;DR

This paper develops a method to approximate functions in the $GSBD^p$ space, crucial for fracture modeling, using piecewise affine finite elements that converge in strain and jump set area.

## Contribution

It introduces a novel approximation technique for $GSBD^p$ functions using Lipschitz continuous functions with controlled jump sets, facilitating analysis in fracture mechanics.

## Key findings

- Strong convergence of strains in $L^p$
- Jump set areas converge to the original
- Approximation via piecewise affine functions

## Abstract

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSBD^p(\Omega)$, $p\in(1,\infty)$, 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 $p\in(1,\infty)$, 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 their 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 Freudhental partition of a cubic grid.