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Existence and convergence for quasi-static evolution in brittle fracture. (English) Zbl 1068.74056
Summary: This paper investigates the mathematical well-posedness of the variational model of quasi-static growth for a brittle crack proposed by G. A. Francfort and J.-J. Marigo [J. Mech. Phys. Solids 46, No. 8, 1319–1342 (1998; Zbl 0966.74060)]. The starting point is a time-discretized version of that evolution which results in a sequence of minimization problems of Mumford-Shah-type functionals. The natural weak setting is that of special functions of bounded variation, and the main difficulty in showing existence of time-continuous, quasi-static growth is to pass to the limit as the time discretization step tends to $$0$$. This is performed with the help of a jump transfer theorem that permits, under weak convergence assumptions on a sequence $$\{u_n\}$$ of SBV functions to its BV limit $$u$$, the transference of the part of the jump set of any test field that lies in the jump set of $$u$$ onto that of the converging sequence $$\{u_n\}$$. In particular, it is shown that the notion of minimizer of a Mumford-Shah-type functional for its own jump set is stable under weak convergence assumptions. Furthermore, our analysis justifies numerical methods used for computing the time-continuous, quasi-static evolution.

##### MSC:
 74R10 Brittle fracture 74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) 74G65 Energy minimization in equilibrium problems in solid mechanics
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