A density result for GSBD and its application to the approximation of brittle fracture energies

Abstract

We present an approximation result for functions u : Ω → R n belonging to the space GSBD(Ω) ∩ L 2 (Ω, R n) with e(u) square integrable and H n−1 (Ju) finite. The approximating functions u k are piecewise continuous functions such that u k → u in L 2 (Ω, R n), e(u k) → e(u) in L 2 (Ω, M n×n sym), H n−1 (Ju k Ju) → 0, and´J u k ∪Ju |u ± k − u ± | ∧ 1dH n−1 → 0. As an application, we provide the extension to the vector-valued case of the Γ-convergence result in SBV (Ω) proved by Ambrosio and Tortorelli in [4, 5].

A Derivation of Linearized Griffith Energies from Nonlinear Models

“…Moreover, we will investigate the limiting model which appears to be more general than many other Griffith functionals in the realm of linearized elasticity (cf. for example [7,13,14,34,43]) as the limiting configuration not only consists of a displacement field, but also of a coarse partition of the domain and associated rigid motions. In particular, it will turn out that there are various scales for the size of the crack opening occurring in the system.…”

“…(Note that indeed the linearized rigidity estimate can also be applied in the GSBD-setting as it relies on a slicing argument and an approximation which is also available in the generalized framework, see [34,Section 3.3]. The only difference is that the approximation does not converge in L 1 but only pointwise almost everywhere, which does not affect the argument.…”

“…For the sake of a simplified mathematical description the investigation of fracture models in the realm of linearized elasticity is widely adopted (see for example [2,4,7,13,14,34]) and has led to a lot of realistic applications in engineering as well as to efficient numerical approximation schemes (we refer to [5,6,12,26,38,39,43] making no claim to be exhaustive). On the contrary, their nonlinear counterparts are usually significantly more difficult to treat since in the regime of finite elasticity the energy density of the elastic contributions is genuinely geometrically nonlinear due to frame indifference rendering the problem highly non-convex.…”

Asymptotic Analysis of Ambrosio--Tortorelli Energies in Linearized Elasticity

“…The quoted result has been later extended in several directions with different aims: to approximate energies arising in the theory of nematic liquid crystals [9], general free discontinuity functionals defined over vector-valued fields [24,25], the Blake and Zisserman second order model in computer vision [5], or fracture models for brittle linearly elastic materials [16,17,29]; to provide a common framework for curve evolution and image segmentation [33,1,2]; to study the asymptotic behavior of gradient damage models under different regimes [22,28]; and to give a regularization of variational models for plastic slip [7].…”

Some recent results on the convergence of damage to fracture

“…In particular the study of lower semicontinuity and relaxation in BD is still in its beginnings (see [10,29,34,26,12,13,21,9]) and integral representation results have been established only in some special cases (see [25]). Compactness and approximation results with more regular functions have been obtained in [10,14,15,23,31].…”