Calculus of Variations and Geometric Measure Theory

S. Conti - M. Focardi - F. Iurlano

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

created by iurlano on 06 Jun 2017
modified by focardi on 17 Dec 2018


Accepted Paper

Inserted: 6 jun 2017
Last Updated: 17 dec 2018

Year: 2017


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 Freudenthal partition of a cubic grid.