EXISTENCE OF STRONG SOLUTIONS FOR QUASI-STATIC EVOLUTION IN BRITTLE FRACTURE JEAN-FRANC ¸ OIS BABADJIAN AND ALESSANDRO GIACOMINI Abstract. This paper is devoted to prove the existence of strong solutions for a brittle fracture model of quasi-static crack propagation in the two dimensional antiplane setting. As usual, the time continuous evolution is obtained as the limit of a discrete in time evolution by letting the time step tend to zero. The analysis rests on a density lower bound estimate for quasi-minimizers of Mumford-Shah type functionals, under a homogeneous Dirichlet boundary condition on a part of the boundary. In contrast with the previous results, since boundary cracks may be obtained as limits of interior cracks, such a density lower bound has to be established also on balls centered inside the domain but possibly intersecting the Dirichlet boundary. Thanks to a 2D geometrical argument, the discrete in time crack turns out to satisfy a uniform density lower bound which can pass to the limit, leading to the closedness of the continuous in time crack. We also establish better convergence properties of the discrete in time displacement/crack pair towards its time continuous counterpart. Keywords: Free discontinuity problems, brittle fracture, quasi-static evolution, functions of bounded variation, regularity. MSC 2010: 49Q20, 35R35, 74R10, 35J70, 49N60. Contents 1. Introduction 1 2. Preliminaries 6 3. Some regularity results for free discontinuity problems 7 4. Quasi-static crack evolutions 12 5. Regular quasi-static evolutions in the two-dimensional anti-plane setting 14 6. Density lower bound for the jump set of quasi-minimizers under homogeneous Dirichlet boundary conditions 22 6.1. Problems on the unit ball 22 6.2. The density lower bound estimate 25 7. Gradient bound estimate for local minimizers of integral functionals 29 7.1. Some properties of the local minimizer 30 7.2. Gradient bound for minimizers of a non-degenerate auxiliary problem 30 7.3. Proof of Theorem 3.8 34 Acknowledgements 35 References 35 1. Introduction The variational approach to fracture introduced by Francfort and Marigo in [16] (see also [3]) is now a well established theory for quasi-static crack propagation. It rests on Griffith’ original idea that crack propagation is the outcome of the interplay between the elastic energy stored in the material, and the surface energy needed to elongate (or create) a crack. If Ω ⊆ R N is the reference configuration of the body, a configuration of Ω is given by a pair (u, Γ) where Γ ⊆ Ω stands for the crack, and u :Ω \ Γ → R N is the associated displacement. Then (u, Γ) carries a total energy of the form (1.1) E (u, Γ) = E el (u, Γ) + E s (Γ), Date : June 7, 2011. 1
EXISTENCE OF STRONG SOLUTIONS FOR QUASI-STATIC EVOLUTION IN BRITTLE FRACTURE
May 23, 2023
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