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

Conti, Sergio; Focardi, Matteo; Iurlano, Flaviana

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 34.

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The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation GSBD$$(Ω), p ∈ (1, ∞), their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of GSBD$$ 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$$ 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.

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