A new proof of compactness in G(S)BD

Abstract

We prove a compactness result in GBD {\operatorname{GBD}} which also provides a new proof of the compactness theorem in GSBD {\operatorname{GSBD}} , due to Chambolle and Crismale. Our proof is based on a Fréchet–Kolmogorov compactness criterion and does not rely on Korn or Poincaré–Korn inequalities.

Brittle fracture in linearly elastic plates

“…With respect to [8], we notice that in the proof of convergence of minima and minimizers we can not rely on the (higher) integrability of the displacement. Hence, we apply the recent compactness result in GSBD p , p > 1, obtained in [15] (see also [3] for an alternative proof and for the case p = 1).…”

“…We now discuss the convergence of minimizers of the functionals E g ρ . To do this, we recall here the GSBD-compactness result obtained in [15, theorem 1.1] (see also [3]).…”