Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy

Abstract

We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness of the jump set for local minimizers of the Griffith energy.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

Existence of strong solutions to the Dirichlet problem for the Griffith energy

“…This shows existence of strong minimisers for (G).As in the other regularity results, the key point (Theorem 5.6) is a density lower bound for the jump set of minimisers, that now holds for all balls centered on a point in J u , contained in the enlarged domain Ω ′ , with radius small enough (and at a small security distance from ∂(∂ D Ω) if Ω is not of class C 1 , see also Remark 5.4). This is the analogous of [7, Theorem 3.4], while in [27,17,14] the density lower bound is proven only for balls contained in Ω.Following the usual scheme by contradiction, we are led to prove a decay estimate for the Griffith energy of local minimisers in balls with vanishing radius and vanishing (n−1)dimensional density of the jump set. If the balls are contained in Ω this is done as in [14], showing that (local) quasi-minimisers with vanishing jump set on the ball B(0, 1), obtained by blow-up, converge to a local minimiser for the bulk energy.In the case where the balls intersect Ω ′ \ Ω we have a sequence of quasi-minimisers for a problem with a prescribed displacement outside Ω: we then modify (Theorem 4.1) the compactness result for functions with vanishing jump [14, Theorem 4] to include the case…”

“…Weak minimisers of (G) were known to exist under simplifying assumptions: an a priori L ∞ assumption on displacement, for [8] in the SBD space [4]; the connectedness of Γ in [10]; a mild fidelity term in [21]; in dimension 2 in [34] (we mention also [33] and the approximations in [36,32,18,15,31,13,11,20]). The regularity result analogous to De Giorgi, Carriero, Leaci [27] is obtained in [17] in dimension 2 (for more general energies) and in [14] in general dimension (see also [19]), ensuring closedness in Ω of the jump set of weak minimisers, thus existence of strong minimisers for the problem with fidelity term. The present work extends this regularity up to the boundary (that is in Ω ′ ), assuming ∂ D Ω of class C 1 , in the main results Theorem 5.7 and Corollary 5.9.…”

An FFT‐based method for computing weighted minimal surfaces in microstructures with applications to the computational homogenization of brittle fracture

“…When this article was written, the author of these lines was not aware of an extension of the work of Braides et al to the case of elasticity, mainly because of reported severe technical challenges. More precisely, as a precursor to homogenization statements, robust existence results need to be provided, see the recent article of Chambolle et al Hossain et al introduced a notion of effective crack resistance that is different from the expression γ eff , cf. Equation , that was established by Braides et al and Cagnetti et al Hossain et al consider mode‐I‐crack propagation in a plane‐stress setting, cf.…”