Volume 25, 2019
|Number of page(s)||12|
|Published online||13 September 2019|
Approximation of fracture energies with p-growth via piecewise affine finite elements
Institut für Angewandte Mathematik, Universität Bonn,
2 DiMaI, Università di Firenze, 50134 Firenze, Italy.
3 Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005 Paris, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 17 March 2018
The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBDp(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 GSBDp 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 Lp 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.
Mathematics Subject Classification: 49J45 / 26B30 / 74R10 / 35A35
Key words: Generalized functions of bounded deformation / p-growth / piecewise finite elements / brittle fracture
© The authors. Published by EDP Sciences, SMAI 2019
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