A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

Abstract

We prove that any function in GSBD p (Ω), with Ω a n-dimensional open bounded set with finite perimeter, is approximated by functions u k ∈ SBV (Ω; R n ) ∩ L ∞ (Ω; R n ) whose jump is a finite union of C 1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies´Ω W (e(u)) dx + H n−1 (Ju), e(u) and Ju being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between u k and u is small in L p outside a sequence of sets E k ⊂ Ω whose measure tends to 0 and if |u| r ∈ L 1 (Ω) with r ∈ (0, p], then |u k −u| r → 0 in L 1 (Ω). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce Γ-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in L 1 (Ω; R n ).

Dynamic perfect plasticity and damage in viscoelastic solids

“…, which follows under the provisions of (20). Eventually, by (19), and (28) this yields the following additional regularity for the displacement, and for the elastic and plastic strains…”

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity

“…We refer to [15,Theorem 1.1]. In contrast to [15], we use here the function ψ(t) := t ∧ 1 for simplicity. It is indeed easy to check that [15, (1.1e)] implies (3.6a).…”

“…We now address the relaxation of F Dir , see (2.5), i.e., a version of F with boundary data. We take advantage of the following approximation result which is obtained by following the lines of [15,Theorem 5.5], where an analogous approximation is proved for Griffith functionals with Dirichlet boundary conditions. The new feature with respect to [15,Theorem 5.5] is that, besides the construction of approximating functions with the correct boundary data, also approximating sets are constructed.…”

“…We take advantage of the following approximation result which is obtained by following the lines of [15,Theorem 5.5], where an analogous approximation is proved for Griffith functionals with Dirichlet boundary conditions. The new feature with respect to [15,Theorem 5.5] is that, besides the construction of approximating functions with the correct boundary data, also approximating sets are constructed. For convenience of the reader, we give a sketch of the proof in Appendix A highlighting the adaptations needed with respect to [15,Theorem 5.5].…”

“…(a) The proof of the lower inequality for the relaxation F is closely related to the analog in [11]: we use an approach by slicing, exploit the lower inequality in one dimension, and a localization method. To prove the upper inequality, it is enough to combine the corresponding upper bound from [11] with a density result for GSBD p (p > 1) functions [15], slightly adapted for our purposes, see Lemma 5.6.…”