Quasistatic evolution of a brittle thin film

Abstract

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L ∞ .

Lower Semicontinuity of Quasi-convex Bulk Energies in SBV and Integral Representation in Dimension Reduction

“…The remainder of the proof is very classical and is essentially the same as that of [4,Lemma 3.6]. As usual, the most delicate point is to prove the subadditivity of E ∞ (u, b, ·), and this is done by gluing together suitable minimizing sequences by means of a cut-off function.…”

“…As pointed out in [4], the main problem with the definition of E in (6.1) is that minimizing sequences are not necessarily bounded in BV (Ω; R 3 ) and thus not necessarily weakly convergent in this space. Following [4], we define for all (u, b, A) ∈ BV (Ω; u, b, A), while we will show that equality holds when u belongs to BV (Ω; R 3 ) ∩ L ∞ (Ω; R 3 ).…”

“…Following [4], we define for all (u, b, A) ∈ BV (Ω; u, b, A), while we will show that equality holds when u belongs to BV (Ω; R 3 ) ∩ L ∞ (Ω; R 3 ). This will be obtained as a consequence of Lemma 6.2.…”

Dimension reduction of a crack evolution problem in a linearly elastic plate

“…Dimension reduction problems for rate-independent processes have been considered only very recently, and to our best knowledge the only related result is due to Babadjian [2] who obtained, by using a different scaling, a two-dimensional evolutionary model of a free crack in a nonlinear elastic membrane.…”

A Quasistatic Evolution Model for the Interaction Between Fracture and Damage

“…This kind of problems has been studied in [38] in the more general framework of stability of rate independent processes through Γ-convergence, which is a well suited mode of convergence for static minimization problems (see [15]). More particular analysis have been carried out for the stability of quasistatic crack evolution through homogenization in [32], or through dimensional reduction in [6].…”