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

¹ Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany.
² DiMaI, Università di Firenze, 50134 Firenze, Italy.
³ Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005 Paris, France.

^* Corresponding author: flavianaiurlano@hotmail.it

Received: 6 October 2017
Accepted: 17 March 2018

Abstract

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBD^p(Ω), p ∈ (1, ∞), 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 ∈ (1, ∞), with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of C¹ 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.