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

Abstract

The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSB{D}^p(\Omega )$, $p\in (1,\infty )$, 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 $GSB{D}^p$ 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.

References in corpus (3)

• Approximation of a Brittle Fracture Energy with a Constraint of Non-Interpenetration
• A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies
• Density of polyhedral partitions

Cited by in corpus (4)

• A density result in $GSB{D}^p$ with applications to the approximation of brittle fracture energies
• Existence of strong solutions to the Dirichlet problem for the Griffith energy
• Equilibrium configurations for epitaxially strained films and material voids in three-dimensional linear elasticity
• On the approximation of $SBD$ functions and some applications