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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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